Question

1. use the remainder theorem to determine which of the following is a factor of the polynomial $p(x)=x^3 + x^2 - 22x - 40$. \\(\bigcirc\\) $(x - 4)$ \\(\bigcirc\\) $(x + 2)$ \\(\bigcirc\\) $(x - 7)$ \\(\bigcirc\\) $(x + 5)$

Explanation:

Step1: Recall the Remainder Theorem

The Remainder Theorem states that if a polynomial \(P(x)\) is divided by \((x - a)\), the remainder is \(P(a)\). If \((x - a)\) is a factor of \(P(x)\), then \(P(a)=0\). For a factor of the form \((x + b)\), we can rewrite it as \((x - (-b))\), so we need to check \(P(-b)=0\).

Step2: Check \((x - 4)\)

For \((x - 4)\), \(a = 4\). Calculate \(P(4)\):

$$
LATEXBLOCK0
$$

So \((x - 4)\) is not a factor.

Step3: Check \((x + 2)\)

For \((x + 2)\), rewrite it as \((x - (-2))\), so \(a=-2\). Calculate \(P(-2)\):

$$
LATEXBLOCK1
$$

Since \(P(-2) = 0\), by the Remainder Theorem, \((x + 2)\) is a factor. (We can stop here, but for completeness, we'll check the other options.)

Step4: Check \((x - 7)\)

For \((x - 7)\), \(a = 7\). Calculate \(P(7)\):

$$
LATEXBLOCK2
$$

So \((x - 7)\) is not a factor.

Step5: Check \((x + 5)\)

For \((x + 5)\), rewrite it as \((x - (-5))\), so \(a=-5\). Calculate \(P(-5)\):

$$
LATEXBLOCK3
$$

So \((x + 5)\) is not a factor.

Answer:

B. \((x + 2)\)