Question

3 a polynomial function is given.
$p(x) = 4x^2 + 9x + 2$
which statement correctly explains whether the binomial $(x - 2)$ is a factor of $p(x)$?
a $(x - 2)$ is a factor of $p(x)$ since $p(0) = 2$.
b $(x - 2)$ is a factor of $p(x)$ since $p(-2) = 0$.
c $(x - 2)$ is not a factor of $p(x)$ since $p(-2) \
eq 0$.
d $(x - 2)$ is not a factor of $p(x)$ since $p(2) \
eq 0$.

Explanation:

To determine if \((x - 2)\) is a factor of \(p(x)\), we use the Factor Theorem. The Factor Theorem states that a binomial \((x - a)\) is a factor of a polynomial \(p(x)\) if and only if \(p(a)=0\).

Step 1: Recall the Factor Theorem

For the binomial \((x - 2)\), we have \(a = 2\). So we need to check the value of \(p(2)\).

Step 2: Calculate \(p(2)\)

Given \(p(x)=4x^{2}+9x + 2\), substitute \(x = 2\) into the polynomial:

$$
LATEXBLOCK0
$$

Since \(p(2)=36
eq0\), by the Factor Theorem, \((x - 2)\) is not a factor of \(p(x)\) because \(p(2)
eq0\).

Now let's analyze each option:

• Option A: \(p(0) = 2\) has nothing to do with whether \((x - 2)\) is a factor. The value of \(p(0)\) relates to the constant term, not the factor \((x - 2)\). So A is incorrect.
• Option B: We calculated \(p(2)=36

eq0\), and \(p(- 2)\) is not relevant for the factor \((x - 2)\) (it would be relevant for \((x + 2)\)). Also, \(p(-2)=4(-2)^{2}+9(-2)+2=16-18 + 2 = 0\), but this is for \((x + 2)\), not \((x - 2)\). So B is incorrect.

• Option C: The value of \(p(-2)\) is relevant for the factor \((x + 2)\), not \((x - 2)\). So C is incorrect.
• Option D: Since \(p(2)

eq0\), by the Factor Theorem, \((x - 2)\) is not a factor of \(p(x)\). So D is correct.

Answer:

D. \((x - 2)\) is not a factor of \(p(x)\) since \(p(2)
eq0\).