High School

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Use long division to rewrite the expression:

[tex]\frac{64 x^3 - 27}{4 x - 3}[/tex]

A. [tex]16 x^2 + 12 x + 9[/tex]
B. [tex]16 x - 7[/tex]
C. [tex]16 x^2 - 12 x + 9[/tex]
D. [tex]64 x^{23} + 48 x + 11[/tex]

Answer :

Let's solve this problem using long division for polynomials, step-by-step.

We want to divide [tex]\(64x^3 - 27\)[/tex] by [tex]\(4x - 3\)[/tex].

Step 1: Setup the Division

Write the division in long division format:

[tex]\[
\begin{array}{c|c}
& 4x - 3 \\
\hline
64x^3 - 27 & \\
\end{array}
\][/tex]

Step 2: Divide the Leading Terms

Divide the first term of the dividend ([tex]\(64x^3\)[/tex]) by the first term of the divisor ([tex]\(4x\)[/tex]):

[tex]\[
\frac{64x^3}{4x} = 16x^2
\][/tex]

This is the first term of the quotient.

Step 3: Multiply and Subtract

Multiply [tex]\(16x^2\)[/tex] by the entire divisor ([tex]\(4x - 3\)[/tex]):

[tex]\[
16x^2 \times (4x - 3) = 64x^3 - 48x^2
\][/tex]

Subtract this from the original polynomial:

[tex]\[
(64x^3 - 27) - (64x^3 - 48x^2) = 48x^2 - 27
\][/tex]

Step 4: Repeat the Process

Now, take the new dividend ([tex]\(48x^2 - 27\)[/tex]).

Divide the first term by the first term of the divisor:

[tex]\[
\frac{48x^2}{4x} = 12x
\][/tex]

This is the next term in the quotient.

Multiply [tex]\(12x\)[/tex] by [tex]\(4x - 3\)[/tex]:

[tex]\[
12x \times (4x - 3) = 48x^2 - 36x
\][/tex]

Subtract to find the new remainder:

[tex]\[
(48x^2 - 27) - (48x^2 - 36x) = 36x - 27
\][/tex]

Step 5: Continue the Process

Divide [tex]\(36x - 27\)[/tex] by [tex]\(4x - 3\)[/tex].

[tex]\[
\frac{36x}{4x} = 9
\][/tex]

Multiply [tex]\(9\)[/tex] by [tex]\(4x - 3\)[/tex]:

[tex]\[
9 \times (4x - 3) = 36x - 27
\][/tex]

Subtract to find the remainder:

[tex]\[
(36x - 27) - (36x - 27) = 0
\][/tex]

Conclusion:

We have completed the division with a remainder of 0. So, the quotient of [tex]\(\frac{64x^3 - 27}{4x - 3}\)[/tex] is [tex]\(16x^2 + 12x + 9\)[/tex].

The correct answer is:

[tex]\(16x^2 + 12x + 9\)[/tex]

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