Accelerated Math II Polynomial Review

Quick Practice “Quiz” 1. A rectangular sheet of metal 36 inches wide is to be made into a trough by turning up sides of equal length (x) perpendicular to the sheet. Find the value of x that will produce the maximum cross sectional area of the trough. What is this maximum cross sectional area?

Quiz Answers 1. The part turned up is x, so the area is A(x) = x(36 – 2x). It has roots at 0 and 18, so the maximum will occur at 9 (the average of the roots.) A(9) = 9(18) = 162, so if we turn up 9”, the trough will have a cross sectional area of 162 square inches.

Quick Practice “Quiz” 2. If F(x) is quadratic, show work to algebraically write the equation for F(x) given the table below.

Quiz Answers 2. If F(x) is quadratic, then F(x) = ax 2 + bx + c, so F(0 ) = 0a + 0b + c = 6 F(1) = 1a + 1b + c = 10 F(2) = 4a + 2b + c = 16 Solve this system of equations to get that a = 1, b = 3, and c = 6. This means F(x) = x 2 + 3x + 6

Quick Practice “Quiz” 3. If f(x) = 2x 3 - x 2 – 26x + 40 A) Determine the number of and nature of the roots of f(x). B) Show all work to algebraically find all roots of f(x). C) Write f(x) in factored form. D) Write all the x intercepts of f(x).

Quiz Answers 3. A) There are 3 complex roots. There are 2 or 0 positive roots and 1 negative root – which means that either 0 or 2 roots are not real roots. The possible rational roots are: ±1, ±2, ±4, ±5, ±8, ±10, ±20, ±40, ±1/2, ±5/2. B) Synthetically divide by 2, and you’re left with 2x 2 + 3x – 20, which factors into (2x – 5)(x + 4), so the roots are 2, 5/2, and -4.

Quick Practice “Quiz” 3. (cont.) If f(x) = 2x 3 - x 2 – 26x + 40 E) Write all the y intercepts of f(x). F) Use all this information to sketch the graph of f(x). G) Write the equation of f(x) if we move its graph to the left 2 and down 3 units.

Quiz Answers 3. C) (x – 2)(x + 4)(2x – 5) D) 2, -4, 5/2 E) 40 F) Check using your calculator. G) 2(x + 2) 3 – (x + 2) 2 – 26(x + 2)

Quick Practice “Quiz” 4. Factor: A) f(x) = x Compare and contrast the graphs of f(x) = x and g(x) = (x - 27) 3. Include comments on the total number of roots and how the graphs show them.

Quiz Answers 4. (x – 3)(x 2 + 3x + 9) 5. f(x) = x 3 – 27 has 3 roots, one positive root at x = 3, and no negative roots. The other two roots are not real. g(x) = (x – 27) 3 has one repeated root at x = 27. f(x) is the graph of x 3 moved down 27 – so the graph only crosses the x-axis once, but g(x) is the graph of x 3 shifted to the right 27, so it curves when it crosses the x-axis – and it is a cubic curve because the root is repeated 3 times.

Quick Practice “Quiz” 6. Write a quartic polynomial with integral coefficients if two of the roots are at 2 - 3i and 1 +  2 A) in factored form. B) in expanded form. 7. How many x intercepts does this quartic have?

Quiz Answers 7. It has 2 x-intercepts because two of the roots are real, but the other 2 aren’t.

Quick Practice “Quiz” 8. Factor: –A) x 2 - 6x y 2 –B) x y 3 –C) x 3 + 7x 2 - 9x - 63 –D) x x x –E) x x + 20

Quick Practice “Quiz” 8. Factor: –A) x 2 - 6x y 2 = (x – 3 + y)(x – 3 – y) –B) x y 3 = (x + 5y)(x 2 – 5xy + 25y 2 ) –C) x 3 + 7x 2 - 9x – 63 = (x + 7)(x – 3)(x + 3) –D) x x x = x(x – 20)(x – 1) –E) x x + 20 = (x – 4)(x – 1)(x + 5)

Quick Practice “Quiz” 9. One of the roots of: p(x) = x 3 - 4x x + 30 is 2. Find the product of the other two roots.

Quick Practice “Quiz” 9. p(x) = x 3 - 4x x + 30 = (x – 2)(x – a)(x – b), so -2ab must equal 30, meaning that the product of the other 2 roots is -15.