Answer :
To solve the problem using long division, we need to divide the polynomial [tex]\(5x^5 + 90x^2 - 135x\)[/tex] by [tex]\(x + 3\)[/tex]. Let's go through the division process step by step:
1. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{5x^5}{x} = 5x^4
\][/tex]
Write [tex]\(5x^4\)[/tex] above the division line as the first term of the quotient.
2. Multiply the entire divisor [tex]\(x + 3\)[/tex] by this term [tex]\(5x^4\)[/tex]:
[tex]\[
(5x^4)(x + 3) = 5x^5 + 15x^4
\][/tex]
3. Subtract this result from the original polynomial:
[tex]\[
(5x^5 + 90x^2 - 135x) - (5x^5 + 15x^4) = -15x^4 + 90x^2 - 135x
\][/tex]
4. Repeat the process for the new polynomial [tex]\(-15x^4 + 90x^2 - 135x\)[/tex]:
[tex]\[
\frac{-15x^4}{x} = -15x^3
\][/tex]
Write [tex]\(-15x^3\)[/tex] next in the quotient.
5. Multiply the divisor by [tex]\(-15x^3\)[/tex]:
[tex]\[
(-15x^3)(x + 3) = -15x^4 - 45x^3
\][/tex]
6. Subtract to find the new polynomial:
[tex]\[
(-15x^4 + 90x^2 - 135x) - (-15x^4 - 45x^3) = 45x^3 + 90x^2 - 135x
\][/tex]
7. Continue with this new polynomial:
[tex]\[
\frac{45x^3}{x} = 45x^2
\][/tex]
Add [tex]\(45x^2\)[/tex] to the quotient.
8. Multiply the divisor by [tex]\(45x^2\)[/tex]:
[tex]\[
(45x^2)(x + 3) = 45x^3 + 135x^2
\][/tex]
9. Subtract to find the new polynomial:
[tex]\[
(45x^3 + 90x^2 - 135x) - (45x^3 + 135x^2) = -45x^2 - 135x
\][/tex]
10. Carry on with the division process:
[tex]\[
\frac{-45x^2}{x} = -45x
\][/tex]
Append [tex]\(-45x\)[/tex] to the quotient.
11. Multiply the divisor by [tex]\(-45x\)[/tex]:
[tex]\[
(-45x)(x + 3) = -45x^2 - 135x
\][/tex]
12. Subtract to find the remainder:
[tex]\[
(-45x^2 - 135x) - (-45x^2 - 135x) = 0
\][/tex]
Since the remainder is 0, the division is complete. The quotient is:
[tex]\[
5x^4 - 15x^3 + 45x^2 - 45x
\][/tex]
Therefore, the correct answer is:
C. [tex]\(5x^4 - 15x^3 + 45x^2 - 45x\)[/tex]
1. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{5x^5}{x} = 5x^4
\][/tex]
Write [tex]\(5x^4\)[/tex] above the division line as the first term of the quotient.
2. Multiply the entire divisor [tex]\(x + 3\)[/tex] by this term [tex]\(5x^4\)[/tex]:
[tex]\[
(5x^4)(x + 3) = 5x^5 + 15x^4
\][/tex]
3. Subtract this result from the original polynomial:
[tex]\[
(5x^5 + 90x^2 - 135x) - (5x^5 + 15x^4) = -15x^4 + 90x^2 - 135x
\][/tex]
4. Repeat the process for the new polynomial [tex]\(-15x^4 + 90x^2 - 135x\)[/tex]:
[tex]\[
\frac{-15x^4}{x} = -15x^3
\][/tex]
Write [tex]\(-15x^3\)[/tex] next in the quotient.
5. Multiply the divisor by [tex]\(-15x^3\)[/tex]:
[tex]\[
(-15x^3)(x + 3) = -15x^4 - 45x^3
\][/tex]
6. Subtract to find the new polynomial:
[tex]\[
(-15x^4 + 90x^2 - 135x) - (-15x^4 - 45x^3) = 45x^3 + 90x^2 - 135x
\][/tex]
7. Continue with this new polynomial:
[tex]\[
\frac{45x^3}{x} = 45x^2
\][/tex]
Add [tex]\(45x^2\)[/tex] to the quotient.
8. Multiply the divisor by [tex]\(45x^2\)[/tex]:
[tex]\[
(45x^2)(x + 3) = 45x^3 + 135x^2
\][/tex]
9. Subtract to find the new polynomial:
[tex]\[
(45x^3 + 90x^2 - 135x) - (45x^3 + 135x^2) = -45x^2 - 135x
\][/tex]
10. Carry on with the division process:
[tex]\[
\frac{-45x^2}{x} = -45x
\][/tex]
Append [tex]\(-45x\)[/tex] to the quotient.
11. Multiply the divisor by [tex]\(-45x\)[/tex]:
[tex]\[
(-45x)(x + 3) = -45x^2 - 135x
\][/tex]
12. Subtract to find the remainder:
[tex]\[
(-45x^2 - 135x) - (-45x^2 - 135x) = 0
\][/tex]
Since the remainder is 0, the division is complete. The quotient is:
[tex]\[
5x^4 - 15x^3 + 45x^2 - 45x
\][/tex]
Therefore, the correct answer is:
C. [tex]\(5x^4 - 15x^3 + 45x^2 - 45x\)[/tex]
