For a third-degree polynomial p(x), the expression x^2 - 4x + 4 is a factor. When p(x) is … — SAT

For a third-degree polynomial $p(x)$ , the expression $x^2 - 4x + 4$ is a factor. When $p(x)$ is divided by $x - 3$ , the remainder is $5$ , and when $p(x)$ is divided by $x + 1$ , the remainder is $-27$ . What is the remainder when $p(x)$ is divided by $x - 1$ ?

Answer: 1

Answer

The remainder when the polynomial is divided by

x - 1

1

The correct answer is

1

. The polynomial is expressed as

p(x) = (x-2)^2(ax + b)

since it is a third-degree polynomial with a factor of

x^2 - 4x + 4 = (x-2)^2

. Applying the Remainder Theorem, we evaluate the polynomial at

x = 3

and

x = -1

, giving the system of equations

3a + b = 5

and

-a + b = -3

. Solving this system yields

a = 2

and

b = -1

, meaning the polynomial is

p(x) = (x-2)^2(2x - 1)

. Finally, the remainder when

p(x)

is divided by

x - 1

p(1) = (1-2)^2(2(1) - 1) = 1

Step-by-Step Solution

Express the third-degree polynomial in terms of its known quadratic factor.

p(x) = (x-2)^2(ax + b)

Since the polynomial is of degree 3 and has a quadratic factor of

x^2 - 4x + 4 = (x-2)^2

, the remaining factor must be linear, of the form

ax+b

Apply the Remainder Theorem to set up the system of equations.

p(3) = 5

and

p(-1) = -27

The Remainder Theorem states that the remainder of a polynomial

p(x)

when divided by

x - c

is equal to

p(c)

Substitute the values of

x

into the polynomial expression.

3a + b = 5

and

9(-a + b) = -27

Evaluating

p(3) = (3-2)^2(3a + b) = 5

yields

3a + b = 5

, and evaluating

p(-1) = (-1-2)^2(-a + b) = -27

yields

9(-a + b) = -27

Simplify the second equation and solve the system of linear equations.

a = 2

and

b = -1

Dividing the second equation by

9

gives

-a + b = -3

. Subtracting this from the first equation (

3a + b = 5

) gives

4a = 8

, which means

a = 2

. Substituting

a = 2

into the first equation yields

b = -1

Calculate the remainder when

p(x)

is divided by

x - 1

p(1) = 1

According to the Remainder Theorem, the remainder of

p(x)

divided by

x - 1

p(1)

. Substituting

x = 1

into

p(x) = (x-2)^2(2x - 1)

yields

(1-2)^2(2(1) - 1) = 1

Key Concept

Using the Remainder Theorem and known factors of a polynomial to solve for unknown coefficients.

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