When the polynomial g(x) = x^4 - 2x^3 + ax^2 - 8 is divided by x - 3, the remainder is 37,… — SAT

When the polynomial $g(x) = x^4 - 2x^3 + ax^2 - 8$ is divided by $x - 3$ , the remainder is $37$ , where $a$ is a constant. What is the value of $a$ ?

Answer: 2

Answer

According to the Polynomial Remainder Theorem, dividing a polynomial

g(x)

by a linear divisor

x - c

yields a remainder equal to

g(c)

. In this problem, the divisor is

x - 3

, so we evaluate the polynomial at

x = 3

and set it equal to the given remainder of

37

. Substituting

3

for

x

g(x) = x^4 - 2x^3 + ax^2 - 8

yields

3^4 - 2(3)^3 + a(3)^2 - 8 = 37

. Simplifying the numerical expressions gives

81 - 54 + 9a - 8 = 37

, which simplifies to

19 + 9a = 37

. Subtracting

19

from both sides results in

9a = 18

. Dividing both sides by

9

gives the value of the constant

a

2

Step-by-Step Solution

Apply the Polynomial Remainder Theorem

g(3) = 37

The Remainder Theorem states that when a polynomial

g(x)

is divided by

x - c

, the remainder is

g(c)

Substitute

x = 3

into the polynomial

g(x)

3^4 - 2(3)^3 + a(3)^2 - 8 = 37

This sets the value of the polynomial evaluated at

x = 3

equal to the given remainder of

37

Simplify the constant terms

19 + 9a = 37

Evaluating the exponents and multiplying:

3^4 = 81

2(3^3) = 54

, and

a(3^2) = 9a

. Combining the constant terms gives

81 - 54 - 8 = 19

Solve the linear equation for

a

a = 2

Subtracting

19

from both sides gives

9a = 18

, and dividing both sides by

9

gives

a = 2

Key Concept

Polynomial Remainder Theorem

Question

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