# 5.3. Polynomials and Polynomial Functions

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1 5.3 Polynomials and Polynomial Functions

2 Polynomial Vocabulary Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a number Polynomial a sum of terms involving variables raised to a whole number exponent, with no variables appearing in any denominator.

3 Polynomial Vocabulary In the polynomial 7x 5 + x 2 y 2 4xy + 7 There are 4 terms: 7x 5, x 2 y 2, -4xy and 7. The coefficient of term 7x 5 is 7, of term x 2 y 2 is 1, of term 4xy is 4 and of term 7 is 7. 7 is a constant term.

4 Types of Polynomials Monomial is a polynomial with one term. Binomial is a polynomial with two terms. Trinomial is a polynomial with three terms.

5 Degrees Degree of a term The degree of a term is the sum of the exponents on the variables contained in the term. Degree of a constant is 0. Degree of the term 5a 4 b 3 c is 8 (remember that c can be written as c 1 ).

6 Degrees Degree of a polynomial The degree of a polynomial is the greatest degree of all its terms. Degree of 9x 3 4x is 3.

7 Evaluating Polynomials Evaluating a polynomial for a particular value involves replacing the value for the variable(s) involved. : Find the value of 2x 3 3x + 4 when x = 2. 2x 3 3x + 4 = 2( 2) 3 3( 2) + 4 = 2( 8) = 6

8 Combining Like Terms Like terms are terms that contain exactly the same variables raised to exactly the same powers. Warning! Only like terms can be combined through addition and subtraction. : Combine like terms to simplify. x 2 y + xy y + 10x 2 y 2y + xy = x 2 y + 10x 2 y + xy + xy y 2y (Like terms are grouped = (1 + 10)x 2 y + (1 + 1)xy + ( 1 2)y = together) 11x 2 y + 2xy 3y

10 Add: (3x 8) + (4x 2 3x + 3). (3x 8) + (4x 2 3x + 3) = 3x 8 + 4x 2 3x + 3 = 4x 2 + 3x 3x = 4x 2 5

11 Add: format y 4y 5 and 5y 1 using a vertical 3 2 8y 4 y 5 2 5y y y 6

12 Subtracting Polynomials To Subtract Polynomials To subtract a polynomial, add its opposite.

13 Subtract 4 ( y 4). 4 ( y 4) = 4 + y + 4 = y = y + 8

14 Subtract ( a 2 + 1) (a 2 3) + (5a 2 6a + 7). ( a 2 + 1) (a 2 3) + (5a 2 6a + 7) = a a a 2 6a + 7 = a 2 a 2 + 5a 2 6a = 3a 2 6a + 11

15 Subtract: (2x 8x 7 x) (3x 2x 3) (2x 8x 7 x) (3x 2x 3) (2x 8x 7 x) ( 3x 2x 3) x 8x 7x 3x 2x x 10x 7x 3

16 Subtract: (9a b 7ab 4 ab ) (6b a 3ab 4 10 b ) 9a b 7ab 4ab 6b a 3ab 4 10b a b 4ab 4ab 10b 4

17 Types of Polynomials Using the degree of a polynomial, we can determine what the general shape of the function will be, before we ever graph the function. A polynomial function of degree 1 is a linear function. We have examined the graphs of linear functions in great detail previously in this course and prior courses. A polynomial function of degree 2 is a quadratic function. In general, for the quadratic equation of the form y = ax 2 + bx + c, the graph is a parabola opening up when a > 0, and opening down when a < 0. a > 0 x a < 0 x

18 Types of Polynomials Polynomial functions of degree 3 are cubic functions. Cubic functions have four different forms, depending on the coefficient of the x 3 term. x 3 coefficient is positive x 3 coefficient is negative

19 5.4 Multiplying Polynomials

20 Multiply. a. (3x 2 )( 2x) = (3)( 2)(x 2 x) = 6x 3 b. 4x(4x 3 + 8) 4x 1 3 (4 x ) 4x(8) 4 6x 32 x 2 2 c. 5 x (2x 7x 2) x x 5x x 5x (2 ) ( )(7 ) ( )( 2) x 35x 10x 2

21 Multiply. 3 6 (2 x )(5 x ) 3 6 2(5)( x )( x ) 9 10x 2 x(5x 4) 2x(5 x) 2 x( 4) 2 10x 8 x x (4x 6x 1) x (4 x ) ( 3 x )( 6 x) ( 3x )(1) 12x 18x 3x 4 3 2

22 Multiplying Two Polynomials To multiply any two polynomials, use the distributive property and multiply each term of one polynomial by each term of the other polynomial. Then combine any like terms.

23 Multiply. a. ( y 5)( y 7) ( y 5 )( y 7) y( y 7) 5( y 7) y y y 7 5 y y y y 2 y y b. (2x 4)(7x + 5) = 2x(7x + 5) 4(7x + 5) = 14x x 28x 20 = 14x 2 18x 20

24 Multiply (a + 2)(a 3 3a 2 + 7). (a + 2)(a 3 3a 2 + 7) = a(a 3 3a 2 + 7) + 2(a 3 3a 2 + 7) = a 4 3a 3 + 7a + 2a 3 6a = a 4 a 3 6a 2 + 7a + 14

25 Multiply (2y 2 + 5)(y 2 + 3y + 4) vertically. 2y 6y 8y y 15 3y 4 2y y y y y 20 y y

26 The FOIL Method When multiplying 2 binomials, the distributive property can be easily remembered as the FOIL method. F product of First terms O product of Outside terms I product of Inside terms L product of Last terms

27 Multiply (y 12)(y + 4). (y 12)(y + 4) (y 12)(y + 4) (y 12)(y + 4) Product of First terms is y 2 Product of Outside terms is 4y Product of Inside terms is 12y (y 12)(y + 4) F O I L (y 12)(y + 4) = y 2 + 4y 12y 48 = y 2 8y 48 Product of Last terms is 48

28 Multiply (2x 4)(7x + 5) F (2x 4)(7x + 5) = I O L F 2x(7x) O + 2x(5) I 4(7x) = 14x x 28x 20 L 4(5) = 14x 2 18x 20

29 Multiply (3x 7y)(7x + 2y) (3x 7y)(7x + 2y) = (3x)(7x + 2y) 7y(7x + 2y) = 21x 2 + 6xy 49xy + 14y 2 = 21x 2 43xy + 14y 2

30 Squaring a Binomial A binomial squared is equal to the square of the first term plus or minus twice the product of both terms plus the square of the second term. (a + b) 2 = a 2 + 2ab + b 2 (a b) 2 = a 2 2ab + b 2

31 Multiply. (x + 6) 2 (x + 6) 2 = (x + 6)(x + 6) F O I L = x 2 + 6x + 6x + 36 The inner and outer products are the same. = x x + 36

32 Multiply. a. (12a 3) 2 = (12a) 2 2(12a)(3) + (3) 2 = 144a 2 72a + 9 b. (x + y) 2 = x 2 + 2xy + y 2

33 Multiplying the Sum and Difference of Two Terms Product of the Sum and Difference of Two Terms The product of the sum and difference of two terms is the square of the first term minus the square of the second term. (a + b)(a b) = a 2 b 2

34 Multiply. (2x + 4)(2x 4) = (2x)(2x) + (2x)( 4) + (4)(2x) + (4)( 4) F O I L = 4x 2 + ( 8x) + 8x + ( 16) The inner and outer products cancel. = 4x 2 16

35 Multiply. a. (5a + 3)(5a 3) = (5a) = 25a 2 9 b. (8c + 2d)(8c 2d) = (8c) 2 (2d) 2 = 64c 2 4d 2

36 2 Multiply [3 (2 a b)] [3 (2 )] 2 a b (3)(2 a b) (2 a b) 9 6(2 a b) (2 a b) 9 12a 6 b (2 a) 2(2 a)( b) ( b) a 6b 4a 4ab b 2 2

37 2 Multiply ( x 3)( x 3)( x 9) 2 ( x 3)( x 3 )( x 9) 2 2 ( x 9 )( x 9) ( x 9) x x 18 81

38 Evaluating Polynomials Techniques of multiplying polynomials are often useful when evaluating polynomial functions at polynomial values. If f(x) = 2x 2 + 3x 4, find f(a + 3). We replace the variable x with a + 3 in the polynomial function. f(a + 3) = 2(a + 3) 2 + 3(a + 3) 4 = 2(a 2 + 6a + 9) + 3a = 2a a a = 2a a + 23

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