High School

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------------------------------------------------ Use long division to find the quotient below.

\[ (5x^5 + 90x^2 - 135x) \div (x+3) \]

A. \( 5x^4 - 15x^3 + 45x^2 - 45x \)

B. \( 5x^3 - 15x^2 + 45x - 45 \)

C. \( 5x + 5x^3 - 25x^2 - 45x \)

D. \( 5x^4 + 15x^3 - 45x^2 - 45x \)

Answer :

Final Answer:

The quotient obtained using long division is [tex]5x^4 - 15x^3 + 45x^2 - 45x[/tex](Option B).

Explanation:

Following the steps of long division:

Arrange the polynomials: Write the dividend [tex](5x^5 + 90x^2 - 135x)[/tex] in descending order of exponents above the divisor (x + 3).

Divide the highest order term: Divide the highest term of the dividend [tex](5x^5)[/tex]by the highest term of the divisor (x). This gives 5x^4 as the first term of the quotient.

Multiply and subtract: Multiply the divisor (x + 3) by the newly obtained term [tex](5x^4)[/tex]. Subtract this product from the corresponding terms of the dividend: [tex](5x^5 - 15x^4) + 90x^2 - 135x.[/tex]

Repeat for lower terms: Continue dividing the remaining terms [tex](90x^2 - 135x)[/tex] by the divisor (x + 3), bringing down terms as needed. This gives[tex]-15x^3 + 45x^2[/tex] and -45x as the remaining terms of the quotient.

Therefore, the final quotient is [tex]5x^4 - 15x^3 + 45x^2 - 45x.[/tex]

Option B is the answer.

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