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Chapter 13 Summary Key Terms • polynomial (13.1) • term (13.1) • coefficient (13.1) • monomial (13.1) • binomial (13.1) • trinomial (13.1) • degree of a term (13.1) • degree of a polynomial (13.1) • Zero Product Property (13.4) 13.1 • Converse of Multiplication • square root (13.6) • positive square root (13.6) • principal square root (13.6) • negative square root (13.6) • extract the square root Property of Zero (13.4) • roots (13.4) • difference of two squares (13.5) • perfect square trinomial (13.5) • difference of two cubes (13.5) • sum of two cubes (13.5) (13.6) • radical expression (13.6) • radicand (13.6) • completing the square (13.7) Identifying Characteristics of Polynomial Expressions A polynomial is an expression involving the sum of powers in one or more variables multiplied by coefficients. A polynomial in one variable is the sum of terms of the form axk where a, called the coefficient, is a real number and k is a non-negative integer. In general, a polynomial is of the form a1xk1 a2xk 2 11 . . . anx0 . Each of the products in a polynomial is called a term. Polynomials are named according to the number of terms: monomials have exactly 1 term, binomials have exactly 2 terms, and trinomials have exactly 3 terms. The exponent of a term is the degree of the term, and the greatest exponent in a polynomial is the degree of the polynomial. When a polynomial is written in standard form, the terms are written in descending order, with the term of the greatest degree first and ending with the term of the least degree. Example © Carnegie Learning The characteristics of the polynomial 13x3 1 5x 1 9 are shown. 1st term 2nd term 3rd term 13x3 5x 9 Coefficient 13 5 9 Power x3 x 1 x0 Exponent 3 1 0 Term 13 There are 3 terms in this polynomial. Therefore, this polynomial is a trinomial. This trinomial has a degree of 3 because 3 is the greatest degree of the terms in the trinomial. 1019 451445_Ch13_941-1026.indd 1019 30/05/13 5:08 PM 13.1 Adding and Subtracting Polynomial Expressions Polynomials can be added or subtracted by identifying the like terms of the polynomial functions, using the Associative Property to group the like terms together, and combining the like terms to simplify the expression. Example 1 Expression: (7x2 2 2x 1 12) 1 (8x3 1 2x2 2 3x) The like terms are 7x2 and 2x2 and 22x and 23x. The terms 8x3 and 12 are not like terms. (7x2 2 2x 1 12) 1 (8x3 1 2x2 2 3x) 8x3 1 (7x2 1 2x2) 1 (22x 2 3x) 1 12 8x3 1 9x2 2 5x 1 12 Example 2 Expression: (4x4 1 7x2 2 3) 2 (2x2 2 5) The like terms are 7x2 and 2x2 and 23 and 25. The term 4x4 does not have a like term. (4x4 1 7x2 2 3) 2 (2x2 2 5) 4x4 1 (7x2 2 2x2) 1 (23 1 5) 4x4 1 5x2 1 2 13.2 Modeling the Product of Polynomials The product of 2 binomials can be determined by using an area model with algebra tiles. Another way to model the product of 2 binomials is a multiplication table which organizes the two terms of the binomials as factors of multiplication expressions. Example 1 13 x 1 1 1 x x2 x x x x x2 x x x 1 x 1 1 1 © Carnegie Learning (2x 1 1)(x 1 3) (2x 1 1)(x 1 3) 5 2x2 1 7x 1 3 1020 Chapter 13 Polynomials and Quadratics 451445_Ch13_941-1026.indd 1020 30/05/13 5:08 PM Example 2 (9x 2 1)(5x 1 7) • 9x 21 5x 45x2 25x 7 63x 27 (9x 2 1)(5x 1 7) 5 45x2 2 5x 1 63x 2 7 5 45x2 1 58x 2 7 13.2 Using the Distributive Property to Multiply Polynomials The Distributive Property can be used to multiply polynomials. Depending on the number of terms in the polynomials, the Distributive Property may need to be used multiple times. Example (2x2 1 5x 2 10)(x 1 7) (2x2 1 5x 2 10)(x) 1 (2x2 1 5x 2 10)(7) (2x2)(x) 1 (5x)(x) 210(x) 1 (2x2)(7) 1 (5x)(7) 2 10(7) 2x3 1 5x2 2 10x 1 14x2 1 35x 2 70 2x3 1 19x2 1 25x 2 70 13.3 Factoring Polynomials by Determining the Greatest Common Factor Factoring a polynomial means to rewrite the expression as a product of factors. The first step in factoring any polynomial expression is to determine whether or not the expression has a greatest common factor. Example © Carnegie Learning Expression: 12x3 1 4x2 1 16x The greatest common factor is 4x. 13 12x3 1 4x2 1 16x 5 4x(3x2) 1 4x(x) 1 4x(4) 5 4x(3x2 1 x 1 4) Chapter 13 Summary 451445_Ch13_941-1026.indd 1021 1021 30/05/13 5:08 PM 13.3 Factoring Trinomials A quadratic expression can be written in factored form, ax2 1 bx 1 c 5 a(x 2 r1)(x 2 r2), by using an area model with algebra tiles, multiplication tables, or trial and error. Factoring a quadratic expression means to rewrite it as a product of two linear expressions. Example 1 Trinomial: x2 1 3x 1 2 Represent each part of the trinomial as a piece of the area model. Then use the parts to form a rectangle. x2 x x x 1 1 x12 x11 x2 x x x 1 1 The factors of this trinomial are the length and width of the rectangle. Therefore, x2 1 3x 1 2 5 (x 1 1)(x 1 2). Example 2 Trinomial: x2 1 15x 1 54 • x 9 x x2 9x 6 6x 54 13.4 Solving Quadratic Equations Using Factoring The Zero Product Property states that if the product of two or more factors is equal to 0, at least one factor must be equal to 0. The property is also known as the Converse of the Multiplication Property of Zero. This property can be used to solve a quadratic equation. The solutions to a quadratic equation are called roots. To calculate the roots of a quadratic equation using factoring: 13 © Carnegie Learning So, x2 1 15x 1 54 5 (x 1 6)(x 1 9). • Perform transformations so that one side of the equation is equal to 0. • Factor the quadratic expression on the other side of the equation. • Set each factor equal to 0. • Solve the resulting equation for the roots. Check each solution in the original equation. 1022 Chapter 13 Polynomials and Quadratics 451445_Ch13_941-1026.indd 1022 30/05/13 5:08 PM Example Equation: 2x2 1 x 5 6 2x2 1 x 5 6 2x2 1 x 2 6 5 6 2 6 2x2 1 x 2 6 5 0 (2x 2 3)(x 1 2) 5 0 2x 2 3 5 0 or x 1 2 5 0 3 ___ 2x 5 __ 2 2 x 5 1.5 13.4 x 5 22 Connecting the Zeros of a Function to the x-intercepts of a Graph The x-intercepts of the graph of the quadratic function f(x) 5 ax2 1 bx 1 c and the zeros of the function are the same as the roots of the equation ax2 1 bx 1 c 5 0. Example Function: f(x) 5 x2 1 6x 2 55 x2 1 6x 2 55 5 0 (x 1 11)(x 2 5) 5 0 x 1 11 5 0 or x 2 5 5 0 x 5 211 x55 The zeros of the function f(x) 5 x2 1 6x 2 55 are x 5 211 and x 5 5. 13.5 Identifying Special Products of Degree 2 © Carnegie Learning There are special products of degree 2 that have certain characteristics. A perfect square trinomial is a trinomial formed by multiplying a binomial by itself. A perfect square trinomial is in the form a2 1 2ab 1 b2 or a2 2 2ab 1 b2. A binomial is a difference of two squares if it is in the form a2 2 b2 and can be factored as (a 1 b)(a 2 b). Example 1 Expression: 4x2 1 12x 1 9 13 4x2 1 12x 1 9 (2x 1 3)(2x 1 3) (2x 1 3)2 This expression is a perfect square trinomial. Chapter 13 Summary 451445_Ch13_941-1026.indd 1023 1023 30/05/13 5:08 PM Example 2 Expression: x2 2 49y2 x2 2 49y2 x2 2 (7y)2 (x 1 7y)(x 2 7y) This binomial is the difference of two squares. 13.5 Identifying Special Products of Degree 3 There are also special products of degree 3 that have certain characteristics. The difference of two cubes is an expression in the form a3 2 b3 that can be factored as (a 2 b)(a2 1 ab 1 b2). The sum of two cubes is an expression in the form of a3 1 b3 that can be factored as (a 1 b)(a2 2 ab 1 b2). Example 1 Expression: 27x3 2 125 27x3 2 125 (3x)3 2 53 (3x 2 5)( (3x)2 1 (3x)(5) 1 52 ) (3x 2 5)(9x2 1 15x 1 25) Example 2 Expression: 64x3 1 1 64x3 1 1 (4x)3 1 13 (4x 1 1)( (4x)2 2 (4x)(1) 1 12 ) © Carnegie Learning (4x 1 1)(16x2 2 4x 1 1) 13 1024 Chapter 13 Polynomials and Quadratics 451445_Ch13_941-1026.indd 1024 30/05/13 5:08 PM 13.6 Determining Approximate Square Roots of Given Values A number b is a square root of a if b2 5 a. There are 2 square roots for every whole number: a positive square root, which is also called the principal square root, and a negative square root. To determine approximate square roots for given values, determine the perfect square that is closest to, but less than, the given value. Also, determine the perfect square that is closest to, but greater than, the given value. You can use these square roots to approximate the square root of the given number. Example ___ Determine the approximate value of √ 40 . 36 # 40 # 49 ___ ___ ___ √ 36 5 6√40 5 ?√49 5 7 6.32 5 39.69 6.42 5 40.96 ___ The approximate value of √ 40 is 6.3. 13.6 Simplifying Square Roots To simplify a square root, extract any perfect squares within the expression. Example ___ Simplify √ 40 . ___ ______ √ 40 5 √ 4 • 10 __ ___ 5 √ 4 • √ 10 ___ 5 2√10 13.6 Extracting Square Roots to Solve Equations © Carnegie Learning The solution to an equation where one term contains a variable and a constant term that are squared can be determined by extracting the square root. To do so, take the square root of both sides of the equation, and then isolate the variable to determine the value of the variable. Example (x 2 7)2 5 75 13 _______ ___ 5√ 75 √ (x 2 7)2 ___ x 2 7 5 6√75 ___ x 5 7 6 √75 x ¯ 7 6 8.7 x ¯ 15.7 or 21.7 Chapter 13 Summary 451445_Ch13_941-1026.indd 1025 1025 30/05/13 5:08 PM Determining the Roots of a Quadratic Equation by Completing the Square 13.7 For a quadratic function that has zeros but cannot be factored, there exists another method for calculating the zeros of the function or solving the quadratic equation. Completing the square is a process for writing a quadratic expression in vertex form which then allows you to solve for the zeros. b When a function is written in standard form, ax2 1 bx 1 c, the axis of symmetry is x 5 2___ . 2a Example Function: f(x) 5 x2 1 4x 1 1 x2 1 4x 1 1 5 0 x2 1 4x 5 21 x2 1 4x 1 4 5 21 1 4 x2 1 4x 1 4 5 3 (x 1 2)2 5 3 _______ __ 5 6√ 3 √ (x 1 2)2 Check: __ x 1 2 5 6√ 3 __ x 5 22 6 √ 3 x ¯ 20.268, 23.732 __ x 5 20.268 ? (20.268)2 1 4(20.268) 1 1 5 0 ? 0.0718 2 1.072 1 1 5 0 x 5 23.732 ? (23.732)2 1 4(23.732) 1 1 5 0 ? 13.9278 2 14.928 1 1 5 0 20.0002 ¯ 0 20.0002 ¯ 0 The roots are 22 6√ 3 . 4 __ © Carnegie Learning The axis of symmetry is x 5 2 , or x 5 22. 2 13 1026 Chapter 13 Polynomials and Quadratics 451445_Ch13_941-1026.indd 1026 30/05/13 5:08 PM