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Define the polynomial.

April 21, 2026Mathematics

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Step-by-step solution

4 steps

To prove that $(x+2)$ is a factor of the polynomial $P(x) = x^3 + 9x^2 + 26x + 24$ , we use the Factor Theorem. The Factor Theorem states that if $(x-c)$ is a factor of a polynomial $P(x)$ , then $P(c) = 0$ . In this case, $c = -2$ .

Define the polynomial.
Let $P(x) = x^3 + 9x^2 + 26x + 24$ .
Substitute $x = -2$ into the polynomial.
$P(-2) = (-2)^3 + 9(-2)^2 + 26(-2) + 24$
Calculate the value of $P(-2)$ .
$P(-2) = -8 + 9(4) - 52 + 24$ $P(-2) = -8 + 36 - 52 + 24$ $P(-2) = 28 - 52 + 24$ $P(-2) = -24 + 24$ $P(-2) = 0$
Conclude based on the Factor Theorem.
Since $P(-2) = 0$ , by the Factor Theorem, $(x+2)$ is a factor of $x^3 + 9x^2 + 26x + 24$ .