The Leading Coefficient Test

University:
Felician University
Course:
MATH-002 | Basic Math Skills-Algebra
Academic year:

2023
Views:

268

Pages:

4
Author:

Clint W.

The Leading Coefficient Test: f(x) = ax^n + ... When n is EVEN a > 0 (positive) Either UP or DOWN Example: 2x^4 + ... a < 0 (negative) Either DOWN or UP Example: -3x^4 + ... When n is ODD a > 0 (positive) Either DOWN or UP Example: 3x^3 + ... a < 0 (negative) Either UP or DOWN Example: -x^5 + ... Real Zeros of Polynomial Functions If f is a polynomial function and a is a real number, the following statements are equivalent. x = a is a zero of the function f. x = a is a solution of the polynomial equation f(x) = 0. (x - a) is a factor of the polynomial f(x). (a, 0) is an x-intercept of the graph of f. Find the zeros of f(x) = x^2 + 2x - 8 x = -4, x = 2 (x + 4)(x - 2) Zeros Solutions Roots Factors X-Intercepts (x + 4)(x - 2) (-4, 0) (2, 0) Repeated Zeros A factor (x - a)^k, k > 1, yields a repeated zero x = a of multiplicity k. If k is odd, the graph crosses the x-axis at x = a. If k is even, the graph touches the x-axis (but does not cross the x-axis) at x = a. BOUNCES Find the zeros of f(x) = -2x^4 + 2x^2 -2x^2(x^2 - 1) -2x^2(x + 1)(x - 1) x = 0 (Multiplicity 2) BOUNCES x = -1 (Multiplicity 1) CROSSES x = 1 (Multiplicity 1) CROSSES