Question

Use the Factor Theorem to determine whether - 3 is a factor of P(x) = -2x4 + 4x + 5x² +9.Specifically, evaluate P at the proper value, and then determine whether x - 3 is a factor,p() -O x- 3 is a factor of P(x)0 * - 3 is not a factor of P(x)

Answer 1

(a) x-3 is a factor of P(x)

Explanation:

You want to know if x -3 is a factor of P(x) = -2x^4 +4x^3 +5x^2 +9 using the factor theorem.

Factor theorem

The factor theorem tells you that x-a is a factor of P(x) if P(a) = 0. To find out if x -3 is a factor, we need to evaluate P(3).

P(3) = (((-2·3) +4)·3 +5)·3² +9 = ((-2·3 +5)·9 +9 = -9 +9 = 0

The function value is 0 at P(3), so (x -3) is a factor of P(x).

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Additional comment

We like to evaluate polynomials of higher degree using the Horner form. This minimizes the arithmetic, and generally gives smaller numbers to work with. As a simple example, ...

ax² +bx +c = (ax +b)·x +c

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