The polynomial function p is defined by p(x) = x^3 + 4x^2 - 7x - 10. If p(2) = 0, which of… — SAT

The polynomial function $p$ is defined by $p(x) = x^3 + 4x^2 - 7x - 10$ . If $p(2) = 0$ , which of the following expressions must be a factor of $p(x)$ ?

A
$x + 2$
$x - 2$ Answer
C
$x - 1$
D
$x - 5$

Answer

The expression

x - 2

must be a factor of

p(x)

According to the Factor Theorem, if a polynomial

p(x)

evaluates to

0

at a certain value

x = c

(meaning

p(c) = 0

), then

(x - c)

is a factor of the polynomial. Since we are given that

p(2) = 0

, the value

2

is a root of

p(x)

, which means the expression

x - 2

must be a factor of the polynomial.

Step-by-Step Solution

Identify the given root of the polynomial.

The problem states that

p(2) = 0

, which means that

x = 2

is a root of the polynomial

p(x)

A root is any value of

x

for which the polynomial evaluates to

0

Apply the Factor Theorem to write the corresponding factor.

According to the Factor Theorem, if

c

is a root of a polynomial

p(x)

, then

(x - c)

is a factor of

p(x)

. Substituting

c = 2

gives

(x - 2)

as a factor.

The Factor Theorem directly relates the roots of a polynomial to its linear factors.

Key Concept

The Factor Theorem states that a polynomial

p(x)

has a factor

(x - c)

if and only if

p(c) = 0

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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