MAFS Algebra 1 Polynomials Day 15 - Student Packet
Day 15: Polynomials MAFS.91.A-SSE.1., MAFS.91.A-SSE..3a,b, MAFS.91.A-APR..3, MAFS.91.F-IF.3.7c I CAN rewrite algebraic expressions in different equivalent forms using factoring techniques use equivalent forms of a quadratic expression to interpret the expression s terms, factors, zeros, maximum, minimum, coefficients, or parts in terms of the real-world situation the expression represents find the zeros of a polynomial function when the polynomial is in factored form create a rough graph of a polynomial function in factored form by examining the zeros of the function use the x-intercepts of a polynomial function and end behavior to graph the function Multiplying and Factoring Polynomials To multiply two polynomials, multiply each term in the first polynomial by each term in the second. The Distributive Property can be used to simplify the product of two or more polynomials. For example, if each polynomial has two terms, with real numbers a, b, c, and d, then (a b)(c d) (a b)c (a b)d ac bc ad bd. To find the product of two variables raised to a power, use the properties of exponents. If the bases are the same, add the exponents: x n x m x n+m If the bases are not the same, then the exponents cannot be added. Example: x n y m x n y m To find the product of a variable with a coefficient and a numeric quantity, multiply the coefficient by the numeric quantity. If a and b are real numbers, then ax b abx. The product of two polynomials is a polynomial, so the system of polynomials is closed under multiplication. Factoring is the reverse process of multiplication. When factoring, it is always helpful to look for a GCF that can be pulled out of the polynomial expression. For example, ab a can be factored as a(b ). Factor the difference of perfect squares: When squaring a binomial (a b), a b : (a b)(a b). (a b) a ab b. Problem Set 1. Find each product. ( )( ) ( )( ) ( )( ) d. ( )( ). The area of a rectangle is found using the formula, where is the length of the rectangle and is the width. Find the area of each rectangle with the given lengths and widths. 1
3. For each of the following, factor out the greatest common factor: d. e. 4. Multiply. ( )( ) ( )( ) ( ) d. ( ) e. ( ) 5. Factor the following examples of the difference of perfect squares. d. 4 e. f. 4 g. 6. The measure of a side of a square is x units. A new square is formed with each side 6 units longer than the original square s side. Write an expression to represent the area of the new square. (Hint: Draw the new square and count the squares and rectangles.)
7. In the accompanying diagram, the width of the inner rectangle is represented by and the length by. The width of the outer rectangle is represented by and the length by. Write an expression to represent the area of the larger rectangle. Write an expression to represent the area of the smaller rectangle. Express the area of the region inside the larger rectangle but outside the smaller rectangle as a polynomial in terms of. (Hint: You will have to add or subtract polynomials to get your final answer.) 8. Multiply the following binomials; note that every binomial given in the problems below is a polynomial in one variable,, with a degree of one. Write the answers in standard form, which in this case will take the form, where,, and are constants. ( )( ) ( )( ) ( )( ) d. ( ) ( ) e. ( ) ( ) 9. Factor the following quadratic expressions. d. e. f. g. 3
10. Factor completely: 11. Factor completely: 1. Factor completely: 13. Factor completely: 14. Factor these trinomials as the product of two binomials, and check your answer by multiplying. 15. Factor completely. 16. The parking lot at Gene Simon s Donut Palace is going to be enlarged so that there will be an additional 30 ft. of parking space in the front of the lot and an additional 30 ft. of parking space on the side of the lot. Write an expression in terms of x that can be used to represent the area of the new parking lot. x 30 x 30 Explain how your solution is demonstrated in the area model. 4
17. Factor the following quadratic expressions. d. e. f. 18. Factor the following quadratic expressions. [Hint: Look for a GCF first.] d. 3 For Exercises 19 -, use the structure of these expressions to factor completely. 19. 63 0. 1.. 3. Factor the following quadratic expressions. f. g. h. d. i. e. j. 5
4. The area of the rectangle below is represented by the expression square units. Write two expressions to represent the dimensions, if the length is known to be twice the width. 18x 1x Two mathematicians are neighbors. Each owns a separate rectangular plot of land that shares a boundary and has the same dimensions. They agree that each has an area of square units. One mathematician sells his plot to the other. The other wants to put a fence around the perimeter of his new combined plot of land. How many linear units of fencing will he need? Write your answer as an expression in. Note: This question has two correct approaches and two different correct solutions. Can you find them both? Relevant Vocabulary Equivalent Polynomial Expressions: Two polynomial expressions in one variable are equivalent if, whenever a number is substituted into all instances of the variable symbol in both expressions, the numerical expressions created are equal. Terms of a Polynomial: When a polynomial is expressed as a monomial or a sum of monomials, each monomial in the sum is called a term of the polynomial. Like Terms of a Polynomial: Two terms of a polynomial that have the same variable symbols each raised to the same power are called like terms. Standard Form of a Polynomial in One Variable: A polynomial expression with one variable symbol, x, is in standard form if it is expressed as a n x n a n 1 x n 1 a 1 x a 0, where n is a non-negative integer, and a 0, a 1, a,, a n are constant coefficients with a n. A polynomial expression in x that is in standard form is often just called a polynomial in x or a polynomial. The degree of the polynomial in standard form is the highest degree of the terms in the polynomial, namely n. The term a n x n is called the leading term and a n (thought of as a specific number) is called the leading coefficient. The constant term is the value of the numerical expression found by substituting into all the variable symbols of the polynomial, namely a 0. 6
Proving Identities Polynomial Identity: A polynomial identity is a statement that two polynomial expressions are equivalent. For example, (x ) x x for any real number x is a polynomial identity. Identities can be used to expand or factor polynomial expressions. A polynomial identity is a true equation that is often generalized so it can apply to more than one example. Common Polynomial Identities Square of Sums Identity (a b) a ab b (a b) (a b)(a b) a ab ab b a ab b Square of Differences Identity (a b) a ab b (a b) (a b)(a b) a ab ab b a ab b Difference of Two Squares Identity a b (a b)(a b) a b a ab ab b (a b)(a b) Sum of Two Cubes Identity a 3 b 3 (a b)( a ab b ) a 3 b 3 Difference of Two Cubes Identity a 3 b 3 (a b)( a ab b ) a 3 b 3 a 3 a b ab a b ab b 3 (a b)( a ab b ) a 3 a b ab a b ab b 3 (a b)( a ab b ) 5. Use polynomial identities to expand or factor each expression. ( ) ( ) d. e. f. ( ) g. h. 7
The Zero Product Property When solving for the variable in a quadratic equation, rewrite the equation as a factored quadratic set equal to zero. Using the zero product property, you know that if one factor is equal to zero, then the product of all factors is equal to zero. Going one step further, when you have set each binomial factor equal to zero and have solved for the variable, all of the possible solutions for the equation have been found. Given the context, some solutions may not be viable, so be sure to determine if each possible solution is appropriate for the problem. Example: Find values of c and d that satisfy each of the following equations. 1. cd Either c or d must be zero, but the other can be any number, including zero (i.e., both c and d MIGHT be equal to zero at the same time).. (c )d There are an infinite number of correct combinations of c and d, but each choice of c will lead to only one choice for d and vice vers For example, if d, then c must be, and if c, then d must be. 3. (c )d Since the product must be zero, there are only two possible solution scenarios that will make the equation true, c can be anything) or d (and c can be anything); specifically, one solution would be c and d. (and d 4. (c 5)(d 3) 0 c or d. Either makes the product equal zero; they could both be true, but both do not have to be true. However, at least one must be true. 5. x x 3 Rewrite the equation in factored form so that is equal zero and solve: (x )(x ), which leads to x or. 6. Solve the following equations. d. 8
Graphing Factored Polynomials Zeros or Roots of a Function: A zero (or root) of a function f: R R is a number x of the domain such that f(x) zero of a function is an element in the solution set of the equation f(x).. A Given any two polynomial functions p and q, the solution set of the equation p(x)q(x) the two equations p(x) and q(x) and combining the solutions into one set. can be quickly found by solving The x-intercepts in the graph of a function correspond to the solutions to the equation f(x) and correspond to the number of distinct zeros of the function (but the x-intercepts do not help us to determine the multiplicity of a given zero). The graph of a polynomial function of degree n has at most n x-intercepts but may have fewer. A polynomial function whose graph has m x-intercepts is at least a degree m polynomial. A polynomial of degree n may have up to n 1 relative maximum/minimum points. A relative maximum is the x-value c that produces the highest point on a graph of f in a circle around c, f(c). That highest value f(c) is a relative maximum value. A relative minimum is the x-value d that produces the lowest point on a graph of f in a circle around d, f(d). That lowest value f(d) is a relative minimum value. End Behavior To determine the end behavior of a polynomial function, or the behavior of the graph as x approaches positive or negative infinity, consider the highest degree of the polynomial and its coefficient, ax n. If n is even, the polynomial function is considered an even-degree polynomial function. If n is odd, the polynomial function is considered an odd-degree polynomial function. Even-degree Polynomials Positive leading coefficient Example: y x Negative leading coefficient Example: y x Odd-degree Polynomials Positive leading coefficient Example: y x 3 Negative leading coefficient Example: y x 3 9
7. Sketch a graph of the function ( ) ( )( )( ) by finding the intercepts and determining the sign of the function between zeros. Explain how the structure of the equation helps guide your sketch. 8. Sketch a graph of the function ( ) ( )( )( ) by finding the zeros and determining the sign of the values of the function between zeros. 9. A function has zeros at,, and. We know that ( ) and ( ) are negative, while ( ) and ( ) are positive. Sketch a graph of. 10
30. Write everything you know about the following polynomial: Degree of the polynomial: x-intercepts: y-intercept: maximum: minimum: Interval(s) where the function is positive: Interval(s) where the function is negative: Interval(s) where the function is increasing: Interval where the function is decreasing: End behavior: 11