Polynomials 370 UNIT 10 WORKING WITH POLYNOMIALS. The railcars are linked together.

UNIT 10 Working with Polynomials The railcars are linked together. 370 UNIT 10 WORKING WITH POLYNOMIALS

Just as a train is built from linking railcars together, a polynomial is built by bringing terms together and linking them with plus or minus signs. You can perform basic operations on polynomials in the same way that you add, subtract, multiply, and divide numbers. Big Ideas A number is any entity that obeys the laws of arithmetic; all numbers obey the laws of arithmetic. The laws of arithmetic can be used to simplify algebraic expressions. Expressions, equations, and inequalities express relationships between different entities. Unit Topics Working with Monomials and Polynomials Adding and Subtracting Polynomials Multiplying Monomials Multiplying Polynomials by Monomials Multiplying Polynomials The FOIL Method WORKING WITH POLYNOMIALS 371

Working with Monomials and Polynomials Some expressions are monomials or polynomials. Classifying Monomials DEFINITION A monomial is a number, a variable, or the product of a number and one or more variables. The exponents of the variables of monomials must be whole numbers. The degree of a monomial is the sum of the exponents of its variables. Some monomials, called constants, have no variable parts. The degree of a constant term c is equal to zero since c can be written as cx 0. Example 1 Determine whether each expression is a monomial. If it is not, explain why. If it is a monomial, determine its degree. REMEMBER A value raised to the 0 power is equal to 1. Expression Monomial? Degree A. 25 Yes. 0 B. 5x 1 2 No. The variable has an exponent that is not a whole number. C. 3x Yes. 1 D. 3 y 2 No. The variable is in the denominator, so 3 y 2 can be rewritten as 3y 2, and 2 is not a whole number. E. 3x 2 Yes. 2 F. mn 2 p 3 Yes. 6 REMEMBER A term is part of an expression that is added or subtracted. A term can be a number, a variable, or a combination of numbers and variables. Classifying Polynomials by the Number of Terms DEFINITION A polynomial is a monomial or the sum or difference of two or more monomials. (continued) WORKING WITH MONOMIALS AND POLYNOMIALS 373

Each monomial is a term of the polynomial. A polynomial with two terms is a binomial. A polynomial with three terms is a trinomial. The degree of a polynomial is equal to the degree of the monomial with the greatest degree. Example 2 Determine whether each expression is a polynomial. If it is not, explain why. If it is a polynomial, determine its degree. Expression Polynomial? Degree A. 7x Yes. 1 B. 9x 3 yz + 3x 3 + 1 2 Yes. 5 C. 2x x 2 No. The term x 2 has an exponent that is not a whole number. D. a 2 + a No. The term 2 a is not a monomial since it has a variable in the denominator. E. 4 Yes. 0 Writing a Polynomial in Standard Form A polynomial is in standard form when every term is simplified and its terms are listed by decreasing degree. Example 3 Write 21y 3y 2 + 4 + y 3 in standard form. Determine the degree of each term and then list the terms by decreasing degree. 21y 3y 2 + 4 + y 3 degree 1 degree 2 degree 0 degree 3 The polynomial written in standard form is y 3 3y 2 + 21y + 4. Classifying Polynomials You can classify a polynomial by the number of terms it has or by its degree. Polynomials of certain degrees have special names. CLASSIFYING POLYNOMIALS BY THE DEGREE Name Degree Example (in simplified form) Constant 0 10 Linear 1 3x 2 Quadratic 2 x 2 + 4x + 4 Cubic 3 2x 3 + x 2 + 5 Quartic 4 5x 4 + 7x Quintic 5 8x 5 + 6x 4 + 2x 3 + 8x 2 + 3x 11 374 UNIT 10 WORKING WITH POLYNOMIALS

A polynomial of nth degree can be written in the form a n x n + a n 1 x n 1 +... + a 1 x + a 0, where a n denotes the coefficient of the term with a power of n, a n 1 denotes the coefficient of the next term of the polynomial with a power of n 1, and so on. For example, the polynomial 6x 5 + 3x 4 2x 3 + 6x 2 + 3 has the following coefficients. a n a n 1 a n 2 a n 3 a n 4 a n 5 a 5 a 4 a 3 a 2 a 1 a 0 6 3 2 6 0 3 Notice that the coefficient of a term may be the same as another term or may equal zero. Example 4 A. Classify the polynomial 3c + 1 by its degree and number of terms. The polynomial 3c + 1 has two terms, and the greatest degree of its terms is 1. It is a linear binomial. B. Classify the polynomial 8b 3 + 2b 2 5b + 6 by its degree. The degree of the term with the greatest degree is 3, so the polynomial is cubic. Application: Geometry Example 5 The height of a ball thrown into the air can be modeled by the polynomial 16t 2 + v 0 t + h 0. Classify the polynomial in the variable t by its number of terms and by its degree. The polynomial 16t 2 + v 0 t + h 0 has three terms, so it is a trinomial. The degree of the term with the greatest degree is 2, so the polynomial is a quadratic trinomial. Problem Set If the expression is a polynomial, write monomial, binomial, or trinomial to describe it by its number of terms. If the expression is not a polynomial, write not a polynomial. 1. 6 x 2 5x 2. 12 a + 2a 8 3. y 3 4. 5z 2 12z + 16 5. 9 2 x x 6. 2 x 4 y + 4x 2 y 2 WORKING WITH MONOMIALS AND POLYNOMIALS 375

State the degree of the polynomial. If the expression is not a polynomial, write not a polynomial. 7. 8 c 3 8. 3x 2 7x + 10 9. 8d 3 10. 7p 4 2p + 6 11. 11 12. 8x 6y 3 13. y 4 + x 14. 5m 6 n 3 3m 4 n 2 15. 9b 4 c 3 6 b 3 c 4 16. 14x x 1 4 17. 7xy 4x y Write the polynomial in standard form. 18. 2 + 3m 19. 11x + 8x 4 12x 2 20. 6y 2 + 3y 5 10y 21. 9z 2 + 4z 4 z 3 + 2z 23. 5xy 2 + 2x 2 y 5 11x 4 y + 4x 3 y 3 24. 9w 3 z 2 + wz 4 5w 5 z 3 + 8w 2 z 25. 3x 2 y 2 z 3 + 7x 4 yz 5 3xy 4 z 2 + 8 x 6 y 2 z 26. 6a 2 b 4 c 5 + abc + 3a 3 b 3 c 3 + 2a 5 b 2 c 2 8a 4 b 2 c 3 22. 1 2 f 3 + 2f 8f 2 + 12f 4 Solve. 27. What is the degree of the expression that represents the perimeter of this triangle? 2x 2 2x 2 *29. Challenge What is the classification and degree of the expression that represents the difference in perimeter from the large square to the small square? 4x + 5 28. What is the classification and degree of the expression that represents the perimeter of this rectangle? 7s s 2 + 10s 3n + 8 n 2 3 n 2 3 *30. Challenge What is the classification and degree of the expression that represents the sum of the perimeters of the rectangle and triangle? 2x 2 6x 3n + 8 x 3 + 8 x 3 + 8 x 2 + 5 x 2 + 5 x 3 1 2x 2 6x 376 UNIT 10 WORKING WITH POLYNOMIALS

Adding and Subtracting Polynomials You can add and subtract polynomials with like terms. Adding Polynomials REMEMBER Combine like terms to add polynomials. Like terms have the same variables raised to the Example 1 Add. same powers. A. (2a 2 + a + 1) + (3a 6) You can add polynomials vertically or horizontally. Vertically: Align like terms in the same column and add the coefficients of the variables. 2a 2 + a + 1 REMEMBER + 3a 6 To combine like terms, add or 2a 2 + 4a 5 subtract the coefficients and keep the variable part the same. Horizontally: Use the commutative and associative properties to rewrite the sum with like terms grouped together. Then simplify by combining like terms. (2a 2 + a + 1) + (3a 6) = 2a 2 + a + 1 + 3a 6 Associative Property = 2a 2 + (a + 3a) + (1 6) Commutative and Associative Properties of Addition = 2a 2 + 4a + ( 5) Combine like terms. = 2a 2 + 4a 5 Simplify. B. (4x 4 + 2x 3 x 2 + 7) + (10 x 4 + 8x 2 3) Combine the like terms. (4x 4 + 2x 3 x 2 + 7) + ( 10x 4 + 8x 2 3) = 4x 4 + 2x 3 x 2 + 7 + 10x 4 + 8x 2 3 = ( 4x 4 + 10x 4 ) + 2x 3 + ( x 2 + 8x 2 ) + (7 3) = 14x 4 + 2x 3 + 7x 2 + 4 ADDING AND SUBTRACTING POLYNOMIALS 377

TIP Remember to distribute 1 to each term when changing to an addition problem. Subtracting Polynomials Subtraction and addition are inverse operations. To subtract polynomials, you can rewrite the problem as an addition problem. You could also use the distributive property to subtract polynomials. Example 2 Subtract. A. (5x + 6) (x + 2) You can subtract polynomials vertically or horizontally. Vertically: Align like terms in the same column, and rewrite the problem as an addition problem. Subtraction Addition 5x + 6 5x + 6 (x + 2) + ( x 2) 4x + 4 Horizontally: Use the distributive property to remove the parentheses. Then group like terms together and simplify by combining like terms. (5x + 6) (x + 2) = 5x + 6 x 2 Distributive Property = (5x x) + (6 2) Commutative and Associative Properties of Addition = 4x + 4 Combine like terms. B. (12a 2 + 3ab 4b 2 ) ( 8a 2 ab 5b 2 ) (12a 2 + 3ab 4b 2 ) ( 8a 2 ab 5b 2 ) = 12a 2 + 3ab 4b 2 8a 2 + ab + 5b 2 = (12a 2 8a 2 ) + (3ab + ab) + ( 4b 2 + 5b 2 ) = 4a 2 + 4ab + b 2 Application: Geometry Example 3 Use the following triangle to answer the questions. 3x + 1 6x + 2 8x 2 A. Express the perimeter of the triangle as a polynomial. B. Find the perimeter of the triangle when x = 1, 2, and 3.4. 378 UNIT 10 WORKING WITH POLYNOMIALS

A. The perimeter of a triangle is the sum of the lengths of its sides. P = (3x + 1) + (6x + 2) + (8x 2) Write the perimeter as the sum of the side lengths. = 3x + 1 + 6x + 2 + 8x 2 Associative Property = (3x + 6x + 8x) + (1 + 2 2) Commutative and Associative Properties of Addition = 17x + 1 Combine like terms. The perimeter of the triangle is 17x + 1 units. B. Substitute each value of x into the polynomial 17x + 1. x = 1 x = 2 x = 3.4 17x + 1 = 17 1 + 1 17x + 1 = 17 2 + 1 17x + 1 = 17 3.4 + 1 = 17 + 1 = 34 + 1 = 57.8 + 1 = 18 = 35 = 58.8 When x = 1, the When x = 2, the When x = 3.4, the perimeter of the perimeter of the perimeter of the tritriangle is 18 units. triangle is 35 units. angle is 58.8 units. Problem Set Add or subtract. Simplify. 1. (3x 2 + 7) + ( x 2 6x + 4) 2. (5b 2 3b + 2) + (2b 4) 3. (2a 2 a + 5) + ( a 2 + 4a 1) 4. (6y 3 4 y 2 + 7) + ( 3y 3 + y 2 2y 5) 5. (3x 4 2x 3 + 4x 2) + ( 3x 3 2x 2 x + 4) 6. (2a 5 + 3a 2 a) + ( 3a 5 a 4 2a 2 + 3a 4) 7. (3y + 2) (y + 5) 8. (6x 3) (2x 2) 9. (7b + 1) (3b 1) 10. (2a 2 3ab + 5b 2 ) ( a 2 2ab + 3b 2 ) 11. (3a 2 + 7ab 4b 2 ) ( 5a 2 4ab + b 2 ) 12. (6x 2 + xy 12y 2 ) ( 10x 2 + xy 2y 2 ) 13. (3x 2 2xy + 2y 2 ) + ( 2x 2 + 5xy y 2 ) 14. (15x 4 3x 2 + 7) ( 8x 4 5x 3 2x 2 + 4) 15. 2m(2m 5) 3m(m 4) 16. 3x(x 2y + 1) + 2x(4x + y 2) 17. (5x 3 3x 2 + 2x 9) ( 2x 3 4x 2 + 3x 4) 18. (3m 2 2mn + 4n 2 ) + ( 2m 2 + 5mn 6n 2 ) 19. 3p(2p 2 p + 3n 1) 2p(3p 5n + 5) 20. (6a 2 3ab + 2a 4b) ( 2a 2 + 5ab 5a b) Challenge *21. *22. 1 2 (x y 2 ) 2 3 x(x 4) 2y(3y 2) 4a(a + 6) a 3a(2a 5) + a ADDING AND SUBTRACTING POLYNOMIALS 379

Solve. 23. Express the perimeter of the triangle as a polynomial. 8x + 2 5x 4 28. Mica is using this set of figures to make a design. What is the combined perimeter of the figures? Note: All measurements are in millimeters. 9x + 3 24. Express the perimeter of the rectangle as a polynomial. 10y + 2 7y 9 25. The height of Jake s window is 5x 3 inches and the width is 3x + 2 inches. What is the perimeter of Jake s window? 26. Belinda placed stepping stones in the shape of the irregular polygon shown. She will plant thyme around the edge of each stepping stone. What is the total length of planting around each stepping stone? 2x + 4 3x 1 2x + 1 x 2 3 3x x 2 x 2 2x 2 2x 2x + 2 x 2 1 29. Arielle drew an equilateral triangle and a square. She made the sides of the square 2 centimeters less than twice the length of the side of the triangle. Express the combined perimeters of the triangle and square as a polynomial. *30. Challenge Benito s apartment is shown in the floor plan. He plans to put decorative baseboard along the perimeters of the living room, dining room, and bedroom, not including the doorways. Note: All measurements are in feet. x + 2 (all other doorways: x 1) 4x + 2 4x 3 4x Living Room Bath Bedroom 4x + 1 3x + 1 Dining Room Kitchen 4x 2 4x 4 27. Craig measured these three rectangles. 2x 2y 3x + 3y y + 1 A. Express the perimeter of each of the three rooms as a polynomial. B. How many feet of baseboard does Benito need? (Remember to subtract the doorways.) x A B x + 1 C 2x 3 A. Express the perimeter of each rectangle as a polynomial. B. Express the combined perimeters of the three rectangles as a polynomial. 380 UNIT 10 WORKING WITH POLYNOMIALS

Multiplying Monomials Every monomial has a coefficient and most have factors that are powers of variables. Multiplying two monomials means multiplying the coefficients and multiplying the variable powers. When you multiply two powers with the same base, you can use the product of powers property to simplify the product. PRODUCT OF POWERS PROPERTY If a is a real number and m and n are integers, then a m a n = a m+n. Simplifying the Product of Powers Example 1 Use the product of powers property to simplify each product. A. 42 4 5 4 2 4 5 = 4 2+5 Product of Powers Property = 4 7 Simplify. Check 4 2 4 5 4 7 16 1024 16,384 16,384 = 16,384 B. x 3 x x 4 x 3 x x 4 = ( x 3 x 1 ) x 4 Associative Property of Multiplication = x 3+1 x 4 Product of Powers Property = x 4 x 4 Simplify. = x 4+4 Product of Powers Property = x 8 Simplify. MULTIPLYING MONOMIALS 381

Multiplying Monomials To multiply two monomials, use the commutative and associative properties of multiplication to get the constant factors together and each variable power together. Once you have all the like variables together, just use the product of powers property to simplify. Example 2 Find each product. A. 3x 3 2x 5 3x 3 2x 5 = (3 2)( x 3 x 5 ) Commutative and Associative Properties of Multiplication = 6( x 3 x 5 ) Multiply. = 6x 3+5 Product of Powers Property = 6x 8 Simplify. B. 5ab 4 ab 5ab 4 ab = 5( a 1 a 1 )(b 4 b 1 ) Commutative and Associative Properties of Multiplication C. = 5a 1+1 b 4+1 Product of Powers Property = 5a 2 b 5 Simplify. 1 2 x 3 y 2 z 2 x 5 y 1 2 x 3 y 2 z 2 x 5 y = ( 1 2 2 ) (x 3 x 5 )(y 2 y 1 )z Commutative and Associative Properties of Multiplication = 1 x 3+5 y 2+1 z Product of Powers Property = x 8 y 3 z Simplify. Application: Geometry Example 3 Use the following figure to answer the questions. 5.5x 3x A. Express the area of the rectangle as a monomial. B. Find the area when x is 3 in., 5 km, and 7.9 cm. 382 UNIT 10 WORKING WITH POLYNOMIALS

A. The area of a rectangle is the product of its length and width. A = lw = 5.5x 3x Substitute 5.5x for l and 3x for w. = (5.5 3)( x 1 x 1 ) Commutative and Associative Properties of Multiplication = 16.5( x 1 x 1 ) Multiply. = 16.5( x 1+1 ) Product of Powers Property = 16.5x 2 Simplify. The area of the rectangle is 16.5x 2 square units. B. Substitute the given values for x into the monomial 16.5x 2. x = 3 in. x = 5 km x = 7.9 cm 16.5x 2 = 16.5 (3 in.) 2 16.5x 2 = 16.5 (5 km) 2 16.5x 2 = 16.5 (7.9 cm) 2 = 16.5 9 in 2 = 16.5 25 km 2 = 16.5 62.41 cm 2 = 148.5 in 2 = 412.5 km 2 = 1029.765 cm 2 When x is 3 inches, When x is 5 kilometers, When x is 7.9 centimeters, the area of the the area of the rectangle the area of the rectangle rectangle is is 412.5 square is 1029.765 square 148.5 square inches. kilometers. centimeters. Problem Set Multiply and simplify. 1. 3 3 3 8 2. a 2 a 6 3. x 2 x x 5 4. 2 9 2 2 2 4 5. a 2 a 4 a a 6 6. 4y 2 6y 3 7. 3xy 2 x 3 y 8. 4b 3 c 2 7bc 2 9. 2b 8 ( 6b 12 ) 10. 5abc 5 3a 2 b 3 11. 3s s 2 t 6 12. 1 3 m 3 np 4 18mn 7 p 3 13. 6z 3w 2 ( 1 9 z 3 w 5 ) 14. 3a 2 ( b 3 c) a 5 bc 2 15. 2 x 3 yz 6 ( 4x 4 y 7 ) 16. 3m 0.4n 2 ( m 5 n 3 ) 17. 2 3 d c 4 3cd 2 18. 5 6 y x 3 y 2x 4 19. 2x 3 y 7 z 1 3 xy z 5 ( 1 2 x 2 z 2 ) Challenge *20. 1 2 xy 2 z 3 9x 5 y 6 3 *21. 6 a 2 b 3 ( a 3 c) ( 3b 4 c 2 ) + ( 1 2 a 5 b ) 8 b 6 c 3 *22. 3x 2y x x 3 y x 4 MULTIPLYING MONOMIALS 383

Solve. 23. Express the area of the rectangle as a monomial. 4.5x 8x 24. Express the area of the triangle as a monomial. 28. The width of a certain rectangle is equal to 9 times the side length s of a certain cube. The length of the rectangle is equal to the volume of the cube. Express the area of the rectangle as a monomial. *29. Challenge Alex has two apartments for rent, as shown in the outlines below. First floor 9x Second floor 9x 7x 6x 7x 2x 3x 3x 5x 2x 2x 25. Mei drew a triangle with a base of 3.7y and a height of 5.2y. A. Express the area of the triangle as a monomial. 6x B. Find the area of the triangle if y = 5 centimeters. 26. Fernando s rectangular dining room table measures 2.4x meters by 1.6x meters. Express the area of the table as a monomial. 27. A cube has side length 6x. A. Express the volume of the cube as a monomial. B. Find the volume of the cube if x = 2 inches. A. Express the total area of each apartment as a monomial. B. By how many square feet is the area of the first floor apartment greater than the second floor apartment if x = 2? *30. Challenge Mrs. King has a coffee table in the shape of the hexagon shown. What is the area of the table? (Hint: divide the hexagon into triangles.) x x 2 3 x 2x 384 UNIT 10 WORKING WITH POLYNOMIALS

Multiplying Polynomials by Monomials Use the distributive property to multiply a polynomial by a monomial. You have used the distributive property to multiply a binomial by a monomial: a(b + c) = ab + ac. The distributive property is true for any number of terms inside the parentheses. For example, a(b + c + d) = ab + ac + ad. You can also use the distributive property when the order of the factors is reversed. For example, (b + c)a = ba + ca. Multiplying a Polynomial by a Monomial MULTIPLYING A POLYNOMIAL BY A MONOMIAL Step 1 Step 2 Use the distributive property to multiply the monomial by each term of the polynomial. Multiply each set of monomials. Use the product of powers property when necessary. REMEMBER The product of powers property states a m a n = a m+n. Example 1 Find each product. A. 4x(x + 3) 4x(x + 3) = 4x x + 4x 3 Distributive Property = 4 x 2 + 12x Multiply. Use the product of powers property for the first term. B. 7y(3y 3 4y 2 y) 7y(3y 3 4y 2 y) = 7y 3y 3 + 7y ( 4y 2 ) + 7y ( y) Distributive Property = 21y 4 28y 3 7y 2 Multiply. Use the product of powers property. C. (21ab 3 + 5ab 2 3ab 7) a 2 b (21ab 3 + 5ab 2 3ab 7) a 2 b = 21ab 3 a 2 b + 5ab 2 a 2 b 3ab a 2 b 7 a 2 b Distributive Property = 21 a 1 a 2 b 3 b 1 + 5 a 1 a 2 b 2 b 1 3 a 1 a 2 b 1 b 1 7a 2 b 1 Commutative Property of Multiplication = 21a 3 b 4 + 5a 3 b 3 3a 3 b 2 7 a 2 b Product of Powers Property MULTIPLYING POLYNOMIALS BY MONOMIALS 385

Application: Area Example 2 Write a polynomial that represents the area of the shaded region. x 8 x Find the area of the shaded region by subtracting the area of the triangle from the area of the rectangle. Step 1 Step 2 Step 3 Find the area of the triangle. A = 1 2 bh = 1 2 x(x 8) The base of the triangle is x and the height is x 8. = 1 2 x x 2 1 x 8 Distributive Property = 1 2 x 2 4x Simplify. The area of the triangle is 1 2 x 2 4x. Find the area of the rectangle. A = lw = x(x 8) The length of the rectangle is x and the width is x 8. = x x x 8 Distributive Property = x 2 8x Simplify. The area of the rectangle is x 2 8x. Find the area of the shaded region. Area of the shaded region = Area of rectangle Area of triangle = ( x 2 8x) ( 1 2 x 2 4x ) = x 2 8x 1 2 x 2 + 4x Distributive Property = x 2 1 2 x 2 8x + 4x The area of the shaded region is 1 2 x 2 4x. = 1 2 x 2 4x Simplify. Commutative Property of Addition 386 UNIT 10 WORKING WITH POLYNOMIALS

Problem Set Multiply and simplify. 1. x(x + 4) 2. a(a 9) 3. 3x(x 1) 4. 2y(y + 6) 5. 7m(12 m) 6. 8x(6x + 2) 7. 5 z 3 (2z 10) 8. (3x + 7) 8x 2 9. 8n(2n 2 n + 1) 10. a(3a 3 + 2a 2 + a) 11. y(2y 5 + 4y 3 + 5y 2 ) 12. 10w(5w 6 + 4w 3 2w) Solve. 25. A large rectangular box is constructed to hold shipping supplies. Write a polynomial to represent the volume of the box (V = lwh). 13. 4x 3 (6x 10 10x 9 + 9x 2 ) 14. 7a 3 ( 10a 3 18a 2 + a) 15. 12x 3 (3x 9 12x 8 8x 7 ) 16. 7p 6 ( 4p 5 9p 4 7p 3 ) 17. xy(3x 2 y 4xy 2 + 2xy 8) 18. (9x 5 y 4 + 6x 4 y 3 + 7x 3 y 2 + 12) x 2 y 19. 2mn 2 (m 3 n 2 + 7m 2 n + 11mn 20) 20. 5xy 4 (2x 4 y 2 8x 3 y + 14x 2 y + 12) 21. (5a 12 b 2 + 7a 10 b 4 4 a 3 b 2 + 6) 3ab 5 22. (3x 7 y 13x 6 y + 10xy 30)4xy 6 23. a 4 b 3 c 2 (3a 6 b 5c 11a 5 b 4 c 2 13a 4 b 3 c 3 14a 3 b 2 c 4 + ab) 24. 4x 2 y 3 z(11x 5 y 4 z 9x 3 y 2 z 2 10x 2 yz 3 14z 4 + z 3 ) 27. Write a polynomial that represents the area of the shaded region formed by the two triangles. 2b 2 ab a + 5b 26. Luke is 5 years older than Olga and Condi is 4 years younger than Olga. Write a polynomial that represents Olga s age times Luke s age decreased by Condi s age. 2s s + 1 s s + 10 MULTIPLYING POLYNOMIALS BY MONOMIALS 387

28. Write a polynomial that represents the area of the shaded walkway formed by these two rectangles. 12x *30. Challenge A quilt pattern is being constructed below. All diagrams are drawn to scale. A square is cut for the first step. In Step 2, a 1 unit square and a rectangle that is 1 unit wide are added to the original square. Find the area and perimeter of the resulting figure in each step. x 7 x + 6 x x 1 x x *29. Challenge Write a polynomial that represents the area of the region formed by the rectangle and semicircle. Step 1 A = P = Step 2 A = P = 1 1 x + 5 x 388 UNIT 10 WORKING WITH POLYNOMIALS

Multiplying Polynomials Using the distributive property to multiply polynomials is similar to multiplying a polynomial by a monomial. USING THE DISTRIBUTIVE PROPERTY TO MULTIPLY TWO POLYNOMIALS Step 1 Step 2 Step 3 Step 4 Use the distributive property to multiply each term of the first polynomial by the second polynomial. Use the distributive property to multiply the monomials by each term of the polynomial. Multiply each set of monomials. If necessary, combine like terms to simplify. Multiplying a Binomial by a Binomial REMEMBER Example 1 Multiply (3x + 2)(x 5). A binomial is a polynomial with Use the distributive property. Think of (x 5) as a single value. two terms. Multiply 3x by (x 5) and 2 by (x 5). (3x + 2)(x 5) = 3x(x 5) + 2(x 5) Distributive Property = 3x x + 3x ( 5) + 2 x + 2 ( 5) Distributive Property = 3x 2 15x + 2x 10 Multiply the monomials. = 3x 2 13x 10 Combine like terms. Squaring a Binomial When you multiply two binomials and both factors are equivalent, you are squaring the binomial. In the diagram below, each side of the square has a side length of a + b. Use the diagram to see that (a + b) 2 = a 2 + 2ab + b 2. a a + b + a 2 ab b ab b 2 (continued) MULTIPLYING POLYNOMIALS 389

SQUARE OF A BINOMIAL ( a + b) 2 = a 2 + 2ab + b 2 Example 2 Expand (2x + 1) 2. Use the pattern (a + b) 2 = a 2 + 2ab + b 2 to square the binomial. (a + b) 2 = a 2 + 2ab + b 2 (2x + 1) 2 = (2x) 2 + 2 2x 1 + 1 2 Substitute 2x for a and 1 for b. = 2x 2x + 4x + 1 Multiply the monomials. = 4x 2 + 4x + 1 Multiply. Multiplying a Polynomial by a Polynomial Example 3 Find each product. A. (a 4)( 2a 2 + 3a 7) (a 4)(2a 2 + 3a 7) = a(2a 2 + 3a 7) 4(2a 2 + 3a 7) Distributive Property = a 2a 2 + a 3a + a ( 7) 4 2a 2 4 3a 4 ( 7) Distributive Property = 2a 3 + 3a 2 7a 8a 2 12a + 28 Multiply the monomials. = 2a 3 5a 2 19a + 28 Combine like terms. B. ( x 2 + 6x + 9)( 2x 2 x 1) ( x 2 + 6x + 9)(2x 2 x 1) = x 2 (2x 2 x 1) + 6x(2x 2 x 1) + 9(2x 2 x 1) Distributive Property = x 2 2x 2 + x 2 ( x) + x 2 ( 1) + 6x 2x 2 + 6x ( x) + 6x ( 1) + 9 2x 2 + 9 ( x) + 9 ( 1) Distributive Property = 2x 4 x 3 x 2 + 12x 3 6x 2 6x + 18x 2 9x 9 Multiply the monomials. = 2x 4 + 11x 3 + 11x 2 15x 9 Combine like terms. C. (x + 1)(x 1)(5x + 2) First multiply (x + 1)(x 1). Then multiply the product by (5x + 2). Step 1 (x + 1)(x 1) = x(x 1) + 1(x 1) Distributive Property = x x + x ( 1) + 1 x + 1 ( 1) Distributive Property = x 2 x + x 1 Multiply the monomials. = x 2 1 Combine like terms. Step 2 ( x 2 1)(5x + 2) = x 2 (5x + 2) 1(5x + 2) Distributive Property = x 2 5x + x 2 2 1 5x 1 2 Distributive Property = 5x 3 + 2x 2 5x 2 Multiply the monomials. 390 UNIT 10 WORKING WITH POLYNOMIALS

Application: Area Example 4 A. A garden is bordered on three sides by a tiled walkway. Write a polynomial that represents the total area of the garden and the walkway. Fence x ft Garden 10 ft x ft 25 ft Walkway x ft B. If the walkway is 3 feet wide, what is the total area of the garden and the walkway? A. Step 1 Write expressions for the outer length and width of the walkway. The length of the walkway is x + 25 + x = 2x + 25. The width of the walkway is x + 10. Step 2 Use the formula for the area of a rectangle. A = lw = (2x + 25)(x + 10) Substitute (2x + 25) for l and (x + 10) for w. = 2x(x + 10) + 25(x + 10) Distributive Property = 2x 2 + 20x + 25x + 250 Multiply. = 2x 2 + 45x + 250 Combine like terms. The total area of the garden and the walkway is 2x 2 + 45x + 250. B. A = 2x 2 + 45x + 250 Write the expression found in Part A. = 2 3 2 + 45 3 + 250 Substitute 3 for x. = 2 9 + 45 3 + 250 Evaluate 3 2. = 18 + 135 + 250 Multiply. = 403 Add. The total area of the garden and the walkway is 403 square feet. MULTIPLYING POLYNOMIALS 391

Problem Set Multiply and simplify. 1. (x + 4)(x 2) 2. (2u + 1)(u + 1) 3. (4x 7)(x + 5) 4. (6c 9)(2c 4) 5. (5a + 7b)(3a 8b) 6. (x + 2) 2 7. (3y 1) 2 8. (4x + 3) 2 9. (10a 7) 2 10. (3x + 5y) 2 11. (n + 1)( n 2 + 2n + 1) 12. (a 2)( 8a 2 5a + 3) Solve. 25. Write a polynomial that represents the area of a circle with a radius of 7x + 1. 26. Find the volume of the package below. Use the formula V = lwh. 13. (x 10)( 4x 2 3x 8) 14. (3a + 2)( 5a 2 + 10a 9) 15. (6x 5y)(2x 2 7xy 8y 2 ) 16. ( x 2 + 3x + 1)( x 2 + 4x + 2) 17. ( v 2 + 7v 1)( v 2 5v + 3) 18. ( x 2 + 2x 4)( 5x 2 2x 11) 19. (10x 2 + 7x + 11)( 2x 2 5x + 6) 20. (3y 2 + 2y 7)( 4y 2 9y + 3) 21. (z + 3)(z + 5)(z + 1) 22. (x + 2)(x 6)(x 3) 23. (d 4)(d + 12)(2d + 5) 24. (2x + 5)(3x 2)(4x 1) *28. Challenge A swimming pool in the shape of a trapezoid sits on a rectangular deck. Write a polynomial expression for the deck area surrounding the pool. 2 2 4s + 3 x x 2x 10 s + 4 2s 1 27. Given the right triangle below, use the Pythagorean theorem to write a polynomial expression for c 2. 2x + 6 3x 15 *29. Challenge Create a pattern for squaring a trinomial by multiplying (a + b + c) 2. Then use the pattern to multiply (x + 2y + 3z) 2. 2x + 1 c 3x + 11 392 UNIT 10 WORKING WITH POLYNOMIALS

The FOIL Method When you use the distributive property to multiply two binomials, you are multiplying each term in the first binomial by each term in the second binomial. The FOIL method helps you organize the steps used to multiply two binomials. Using the FOIL Method to Multiply a Binomial by a Binomial The letters of the word FOIL stand for First, Outer, Inner, and Last. These words tell you which terms to multiply. First Outer Inner Last (a + b)(c + d) = a c + a d + b c + b d Example 1 Multiply. A. (x + 9)(3x 4) First Outer Inner Last (x + 9)(3x 4) = x 3x + x ( 4) + 9 3x + 9 ( 4) FOIL = 3x 2 4x + 27x 36 Multiply. = 3x 2 + 23x 36 Combine like terms. B. ( a 2 )( a + 4 ) REMEMBER a a = a 2 ( a 2 ) ( First Outer Inner Last a + 4 ) = a a + a 4 + ( 2) a + ( 2) 4 = a + 4 a 2 a 8 = a + (4 2) a 8 = a + 2 a 8 FOIL Multiply. Combine like terms. Simplify. = a THE FOIL METHOD 393

Using the FOIL Method to Multiply Conjugate Binomials Conjugate binomials, (a + b) and (a b), are two binomials with the same terms but opposite signs. CONJUGATE BINOMIALS For any real numbers a and b, (a + b)(a b) = a 2 b 2. Example 2 A. Prove that (a + b)(a b) = a 2 b 2. (a + b)(a b) = a a a b + b a b b FOIL = a 2 ab + ba b 2 Multiply the monomials. = a 2 ab + ab b 2 Commutative Property of Multiplication = a 2 b 2 Additive Inverse Property B. Multiply (5x 7)(5x + 7). Use the pattern (a + b)(a b) = a 2 b 2 to find the product. (a b)(a + b) = a 2 b 2 (5x 7)(5x + 7) = (5x) 2 7 2 Substitute 5x for a and 7 for b. = 5x 5x 7 7 Multiply the monomials. = 25x 2 49 Simplify. (x + y)(x + y) = x x + x y + y x + y y FOIL 394 UNIT 10 WORKING WITH POLYNOMIALS Cubing a Binomial You know a pattern for squaring a binomial. There is also a pattern for cubing a binomial. Example 3 Find the product. A. (x + y) 3 (x + y) 3 = (x + y) (x + y) (x + y) Step 1 Use FOIL to expand (x + y)(x + y). = x 2 + xy + yx + y 2 Multiply the monomials. = x 2 + xy + xy + y 2 Commutative Property of Multiplication = x 2 + 2xy + y 2 Combine like terms. Step 2 Use the distributive property to multiply the product found in Step 1 by (x + y). (x + y)( x 2 + 2xy + y 2 ) = x( x 2 + 2xy + y 2 ) + y( x 2 + 2xy + y 2 ) Distributive Property = x x 2 + x 2xy + x y 2 + y x 2 + y 2xy + y y 2 Distributive Property = x 3 + 2 x 2 y + xy 2 + x 2 y + 2xy 2 + y 3 Multiply the monomials. = x 3 + 3x 2 y + 3xy 2 + y 3 Combine like terms.

The expression (x + y) 3 is a perfect cube, and (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3. You can use this fact to find any perfect cube. B. (a +3) 3 Use the perfect cube pattern: (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3. (x + y) 3 = x 3 + 3x 2 y + 3x y 2 + y 3 (a + 3) 3 = a 3 + 3a 2 3 + 3a 3 2 + 3 3 Substitute a for x and 3 for y. = a 3 + 9a 2 + 27a + 27 Simplify. Problem Set Multiply and simplify. 1. (x + 4)(x + 2) 2. ( j 0.2)(0.13 + j) 3. (d 5)(d 10) 4. (x + 12)(x 3) 5. (q + 1)(q + 5) 6. (v 6)(v 3) 7. (x 7)(x + 3) 8. (a + 11)(a 10) 9. ( x + 7 8 ) ( x 7 8 ) 10. (p 2 )(p 2 ) _ 11. ( z + 2)(18 + z ) 12. (k + 3)(8k + 4) 13. (2a + 1)(a 1) 14. (t 7)(5t 7) Challenge _ *29. A. Find the product of (3x + 1)(x 2). B. Use your work in Part A to find the product of (3x + 1)(x 2)(x + 4). 15. (6.1 + n)(2n + 5) _ 16. ( r 2)(3 r 4) 17. ( 1 4 y + 2 3 ) ( 2 5 y + 1 ) _ 18. (1 3u)(1 + 3u) 19. (3f + 2)(6f + 4) 20. (0.9 2g)(2.5g 3) 21. (20x + 21 )(20x 21 ) 22. (2 y + 7)(1 2 y ) 23. (b + 7)( b 2 2) 24. ( h 2 + 0.3)(1.4 h) 25. ( w 2 + 5)(2w + 1) 26. (6x 2 + 3)(4 x) 27. (x + 6) 3 28. (5 3c) 3 *30. A. Find the product of (x + 4)(x 4). B. Use your work in Part A to find the product of (x + 4) 2 (x 4) 2. THE FOIL METHOD 395