Select the correct answer.

What is the quotient when [tex]\((16x^4 + 40x^3 - 24x^2)\)[/tex] is divided by [tex]\(8x^2\)[/tex]?

A. [tex]\(2x^2 + 3x - 5\)[/tex]
B. [tex]\(2x^3 + 5x^2 - 3\)[/tex]
C. [tex]\(2x^2 - 5x + 5\)[/tex]
D. [tex]\(2x^2 + 5x - 3\)[/tex]



Answer :

To find the quotient when [tex]\(\left(16 x^4 + 40 x^3 - 24 x^2\right)\)[/tex] is divided by [tex]\(8 x^2\)[/tex], we can proceed with polynomial division by performing each term one by one. Here's how you can do it step-by-step:

1. Divide the leading terms of the numerator by the leading term of the divisor:
[tex]\[ \frac{16 x^4}{8 x^2} = 2 x^2 \][/tex]

2. Multiply the entire divisor by this result:
[tex]\[ (2 x^2) \cdot (8 x^2) = 16 x^4 \][/tex]

3. Subtract this result from the original polynomial:
[tex]\[ (16 x^4 + 40 x^3 - 24 x^2) - (16 x^4) = 40 x^3 - 24 x^2 \][/tex]

4. Repeat the process with the new polynomial [tex]\(40 x^3 - 24 x^2\)[/tex]:
[tex]\[ \frac{40 x^3}{8 x^2} = 5 x \][/tex]

5. Multiply the entire divisor by this new result:
[tex]\[ (5 x) \cdot (8 x^2) = 40 x^3 \][/tex]

6. Subtract this result from the current polynomial:
[tex]\[ (40 x^3 - 24 x^2) - (40 x^3) = -24 x^2 \][/tex]

7. Repeat the process with the new term [tex]\(-24 x^2\)[/tex]:
[tex]\[ \frac{-24 x^2}{8 x^2} = -3 \][/tex]

8. Multiply the entire divisor by this new result:
[tex]\[ (-3) \cdot (8 x^2) = -24 x^2 \][/tex]

9. Subtract this result from the current polynomial:
[tex]\[ (-24 x^2) - (-24 x^2) = 0 \][/tex]

So, the quotient is given by the combination of the results from each step:
[tex]\[ \boxed{2 x^2 + 5 x - 3} \][/tex]

Therefore, the correct answer is:
D. [tex]\(2 x^2 + 5 x - 3\)[/tex].