Define the polynomial.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
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, $c = -2$.
Step 1: Define the polynomial.
Let $P(x) = x^3 + 9x^2 + 26x + 24$.
Step 2: Substitute $x = -2$ into the polynomial.
$$P(-2) = (-2)^3 + 9(-2)^2 + 26(-2) + 24$$
Step 3: Calculate the value of $P(-2)$.
$$P(-2) = -8 + 9(4) - 52 + 24$$
$$P(-2) = -8 + 36 - 52 + 24$$
$$P(-2) = 28 - 52 + 24$$
$$P(-2) = -24 + 24$$
$$P(-2) = 0$$
Step 4: Conclude based on the Factor Theorem.
Since $P(-2) = 0$, by the Factor Theorem, $(x+2)$ is a factor of $x^3 + 9x^2 + 26x + 24$.
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