5-3 Study Guide and Intervention

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1 Study Guide and Intervention Polynomial in One Variable A polynomial of degree n in one variable x is an expression of the form a n x n + a n - 1 x n a x + a 1 x + a 0, where the coefficients a n -1, a n -, a n - 3,, a 0 represent real numbers, a n is not zero, and n represents a nonnegative integer. The degree of a polynomial in one variable is the greatest exponent of its variable. The leading coefficient is the coefficient of the term with the highest degree. Polynomial Function A polynomial function of degree n can be described by an equation of the form P(x) = a n x n + a n-1 x n a x + a 1 x + a 0, where the coefficients a n - 1, a n -, a n - 3,, a 0 represent real numbers, a n is not zero, and n represents a nonnegative integer. Example 1 What are the degree and leading coefficient of 3x - x x 3? Rewrite the expression so the powers of x are in decreasing order. -x + x 3 + 3x - 7 This is a polynomial in one variable. The degree is, and the leading coefficient is -. Example Find f(-5) if = x 3 + x - 10x + 0. = x 3 + x - 10x + 0 Original function f(-5) = (-5) 3 + (-5) - 10(-5) + 0 Replace x with -5. = Evaluate. = -5 Simplify. Example 3 g(x) = x + 3x - Find g(a - 1) if g(x) = x + 3x -. Original function g(a - 1) = (a - 1) + 3(a - 1) - Replace x with a - 1. = a - a a Evaluate. = a + a - 6 Simplify. Exercises State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. 1. 3x + 6x 3 - x x x 7 3. x 6 + 6x + 8x 8-10x + 0. x - 3xy + 16y 5. 8x 3-9x 5 + x x 18 - x x Lesson X-3 Find f() and f(-5) for each function. 7. = x = x 3-3x + x = 9x 3 - x + 5x + 7 Chapter 5 17 Glencoe Algebra

2 Study Guide and Intervention Graphs of End Behavior of Polynomial Functions Real Zeros of a Polynomial Function If the degree is even and the leading coefficient is positive, then + as x - + as x + If the degree is even and the leading coefficient is negative, then - as x - - as x + If the degree is odd and the leading coefficient is positive, then - as x - + as x + If the degree is odd and the leading coefficient is negative, then + as x - - as x + (continued) The maximum number of zeros of a polynomial function is equal to the degree of the polynomial. A zero of a function is a point at which the graph intersects the x-axis. On a graph, count the number of real zeros of the function by counting the number of times the graph crosses or touches the x-axis. Example Determine whether the graph represents an odd-degree polynomial or an even-degree polynomial. Then state the number of real zeros. - O x - As x -, - and as x +, +, so it is an odd-degree polynomial function. The graph intersects the x-axis at 1 point, so the function has 1 real zero. Exercises For each graph, a. describe the end behavior, b. determine whether it represents an odd-degree or an even-degree function, and c. state the number of real zeroes O x - - O x - - O x - Chapter 5 18 Glencoe Algebra

3 Skills Practice State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. 1. a + 8. (x - 1)(x + 3) 3. -5x 5 + 3x y + 5y - y 5 + 7y 6 5. u 3 + u t + t 6. r - r + 1 r Find p(-1) and p() for each function. 7. p(x) = - 3x 8. p(x) = 3x + x 9. p(x) = x - x p(x) = -x 3 + 5x p(x) = x + 8x p(x) = 1 3 x - 3 x + If p(x) = x - 3 and r(x) = 1 + 3x, find each value. 13. p(a) 1. r(a) 15. 3r(a) 16. p(a) 17. p(a ) 18. r(x + ) For each graph, a. describe the end behavior, b. determine whether it represents an odd-degree or an even-degree function, and c. state the number of real zeroes O x O x O x - Lesson X-3 Chapter 5 19 Glencoe Algebra

4 Practice State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. 1. (3x + 1)(x - 9). 1 5 a3-3 5 a + 5 a m - 1 m xy3-1x y - 10y Find p(-) and p(3) for each function. 5. p(x) = x 3 - x 5 6. p(x) = -7x + 5x p(x) = -x 5 + x 3 8. p(x) = 3x 3 - x + x p(x) = x + 1 x3-1 x 10. p(x) = 1 3 x x + 3x If p(x) = 3x - and r(x) = x - 5x + 1, find each value. 11. p(8a) 1. r(a ) r(a) 1. r(x + ) 15. p(x - 1) 16. 5p(x + ) For each graph, a. describe the end behavior, b. determine whether it represents an odd-degree or an even-degree function, and c. state the number of real zeroes O x O x O x - 0. WIND CHILL The function C(w) = 0.013w - w - 7 estimates the wind chill temperature C(w) at 0 F for wind speeds w from 5 to 30 miles per hour. Estimate the wind chill temperature at 0 F if the wind speed is 0 miles per hour. Chapter 5 0 Glencoe Algebra

5 Word Problem Practice 1. MANUFACTURING A metal sheet is curved according to the shape of the graph of = x - 9x. What is the degree of this polynomial?. GRAPHS Kendra graphed the polynomial shown below. 6 y. DRILLING The volume of a drill bit can be estimated by the formula for a cone, V = 1 3 πhr, where h is the height of the bit and r is its radius. Substituting 3 3 r for h, the volume of the drill bit is estimated as 3 9 π r3. Graph the function of drill bit volume. Describe the end behavior, degree, and sign of the leading coefficient. V 0 - O 6 x - -6 From this graph, describe the end behavior, degree, and sign of the leading coefficient r 3. PENTAGONAL NUMBERS The nth pentagonal number is given by the expression n(3n - 1). What is the degree of this polynomial? What is the seventh pentagonal number? 5. TRIANGLES Dylan drew n dots on a piece of paper making sure that no line contained 3 of the dots. The number of triangles that can be made using the dots as vertices is equal to f(n) = 1 6 (n3-3n + n). a. What is the degree of f? Lesson b. If Dylan drew 15 dots, how many triangles can be made? Chapter 5 1 Glencoe Algebra

6 Enrichment Approximation by Means of Polynomials Many scientific experiments produce pairs of numbers [x, ] that can be related by a formula. If the pairs form a function, you can fit a polynomial to the pairs in exactly one way. Consider the pairs given by the following table. x We will assume the polynomial is of degree three. Substitute the given values into this expression. = A + B(x - x 0 ) + C(x - x 0 )(x - x 1 ) + D(x - x 0 )(x - x 1 )(x - x ) You will get the system of equations shown below. You can solve this system and use the values for A, B, C, and D to find the desired polynomial. 6 = A 11 = A + B( - 1) = A + B 39 = A + B( - 1) + C( - 1)( - ) = A + 3B + 6C -5 = A + B(7-1) + C(7-1)(7 - ) + D(7-1)(7 - )(7 - ) = A + 6B + 30C + 90D Solve. 1. Solve the system of equations for the values A, B, C, and D.. Find the polynomial that represents the four ordered pairs. Write your answer in the form y = a + bx + cx + dx Find the polynomial that gives the following values. x A scientist measured the volume of carbon dioxide gas that can be absorbed by one cubic centimeter of charcoal at pressure x. Find the values for A, B, C, and D. x Chapter 5 Glencoe Algebra