If the expression \frac{4x^3 - 2x^2 + 7x + 7}{2x^2 + 1} is equivalent to ax + b + \frac{cx… — SAT

If the expression $\frac{4x^3 - 2x^2 + 7x + 7}{2x^2 + 1}$ is equivalent to $ax + b + \frac{cx + d}{2x^2 + 1}$ for all values of $x$ , where $a$ , $b$ , $c$ , and $d$ are constants, what is the value of $a + b + c + d$ ?

Answer: 14

Answer

The value of

a + b + c + d

is 14.

Performing polynomial long division on

\frac{4x^3 - 2x^2 + 7x + 7}{2x^2 + 1}

yields a quotient of

2x - 1

and a remainder of

5x + 8

. Matching this to the form

ax + b + \frac{cx + d}{2x^2 + 1}

gives

a = 2

b = -1

c = 5

, and

d = 8

. Summing these values gives

2 + (-1) + 5 + 8 = 14

Step-by-Step Solution

Divide the leading term of the numerator by the leading term of the denominator to find the first term of the quotient.

\frac{4x^3}{2x^2} = 2x

. Multiplying

2x(2x^2 + 1) = 4x^3 + 2x

. Subtracting this from the numerator yields

-2x^2 + 5x + 7

To initiate the polynomial division process.

Divide the leading term of the remaining polynomial by the leading term of the divisor to find the second term of the quotient.

\frac{-2x^2}{2x^2} = -1

. Multiplying

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

. Subtracting this from the remaining polynomial yields

5x + 8

To find the next term of the quotient and determine the remainder.

Write the expression in the quotient-remainder form and identify the values of the constants

a

b

c

, and

d

The expression is equivalent to

2x - 1 + \frac{5x + 8}{2x^2 + 1}

, so

a = 2

b = -1

c = 5

, and

d = 8

To match the given algebraic form of the expression.

Calculate the sum of the constants

a + b + c + d

2 + (-1) + 5 + 8 = 14

To find the final value requested by the question.

Key Concept

Equivalent Algebraic Expressions

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic