University:
High School
Course:
High School Algebra
Academic year:

2021
Views:

244

Pages:

8
Author:

Giada Roberts

4.5 Solving Polynomial Equations A. VOCABULARY & CONCEPTS Zeros of a function: the x-intercepts. Also called roots, solutions. Rational Number: A number that can be written as a ratio (fraction), decimal terminates or repeats. Irrational Number: A number that cannot be written as a ratio, decimal does not terminate, decimal does not repeat. 1 e.g., π, e, √5 → Any square root that cannot be simplified is irrational Rational Root Theorem: The possible rational roots of a polynomial are: p/q = factors of constant / factors of leading coefficient Irrational Solutions come in pairs. If a + √b is a solution, then its conjugate a - √b is also a solution. e.g., 5 + √3 is a solution, then 5 - √3 is also a solution. B. EXAMPLES 1. Solve the equation a. 2x³ - 12x² + 18x = 0 Factor out common term 2x(x² - 6x + 9) = 0 2x(x - 3)(x - 3) = 0 Use Zero Product Property x = 0 x - 3 = 0 +3 +3 x = 3 b. 5x³ - 40x² + 80x = 0 5x(x² - 8x + 16) = 0 5x(x - 4)(x - 4) = 0 x = 0 x - 4 = 0 +4 +4 x = 4 c. 9m⁵ = 27m³ Let equation equal 0 9m⁵ - 27m³ = 0 9m³(m² - 3) = 0 m³ = 0 m² - 3 = 0 m = 0 m = ±√3 d. 12n² + 48n = -n³ - 64 n³ + 12n² + 48n + 64 = 0 (n + 4)³ = 0 ∛(n + 4)³ = ∛0 n + 4 = 0 -4 -4 n = -4 2. Find the zeros. Then Sketch the graph. a. f(x) = -2x⁴ + 16x² - 32 0 = -2x⁴ + 16x² - 32 Factor to find the Zeros 0 = -2(x⁴ - 8x² + 16) middle exponent is half of leading exponent so can be factored in same way as a quadratic. 0 = -2(x² - 4)(x² - 4) difference of two squares 0 = -2(x + 2)(x - 2)(x + 2)(x - 2) DOUBLE ROOTS means graph "bounces" at the roots (x-intercepts) By ZPP: x + 2 = 0 x - 2 = 0 x = -2 x = 2 zeros (roots/X-intercepts) Graph: degree 4 (quartic) Leading coefficient is negative ↓ y-intercept: y = -2(0)⁴ + 16(0)² - 32 = 0 + 0 - 32 = -32 b. f(x) = x⁴ - 18x² + 81 0 = (x² - 9)(x² - 9) 0 = (x + 3)(x - 3)(x + 3)(x - 3) 0 = (x + 3)²(x - 3)² → graph bounces at x = -3, x = 3 By ZPP: x + 3 = 0 x - 3 = 0 -3 -3 +3 +3 x = -3 x = 3 Zeros (roots/x-intercepts) Graph: degree 4 (quartic) Leading coefficient is positive ↑ graph bounces at x = 3, x = -3 y-intercept: y = (0)⁴ - 18(0)² + 81 = 0 + 0 + 81 = 81