The polynomial function p is defined by p(x) = x^4 - 3x^3 + ax^2 - 7x + 10, where a is a c… — SAT

The polynomial function $p$ is defined by $p(x) = x^4 - 3x^3 + ax^2 - 7x + 10$ , where $a$ is a constant. If $x - 2$ is a factor of $p(x)$ , what is the value of $a$ ?

Answer: 3

Answer

According to the Factor Theorem, if

x - 2

is a factor of the polynomial function

p(x)

, then evaluating the polynomial at

x = 2

must yield a value of

0

. Substituting

x = 2

into the equation gives

p(2) = 2^4 - 3(2)^3 + a(2)^2 - 7(2) + 10 = 0

. Simplifying the terms results in

16 - 24 + 4a - 14 + 10 = 0

, which further simplifies to

4a - 12 = 0

. Adding

12

to both sides and dividing by

4

yields

a = 3

Step-by-Step Solution

Apply the Factor Theorem to set up the equation.

p(2) = 0

By the Factor Theorem,

x - c

is a factor of

p(x)

if and only if

p(c) = 0

Substitute

x = 2

into the polynomial expression.

2^4 - 3(2)^3 + a(2)^2 - 7(2) + 10 = 0

Evaluating

p(2)

yields a linear equation in terms of the unknown constant

a

Simplify the arithmetic terms in the equation.

4a - 12 = 0

Simplifying the constant values:

16 - 24 - 14 + 10 = -12

Solve the linear equation for

a

a = 3

Isolating the variable term gives

4a = 12

, and dividing by

4

yields

a = 3

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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