Answer :

To divide the polynomial [tex]\(-35x^5 - 40x^4 + 15x^3 - 15x^2\)[/tex] by [tex]\(-5x\)[/tex], we can perform polynomial long division or use properties of exponents. Here's how to find the quotient step-by-step:

1. Set Up the Division:
We are dividing each term of the polynomial [tex]\(-35x^5 - 40x^4 + 15x^3 - 15x^2\)[/tex] by [tex]\(-5x\)[/tex].

2. Divide Each Term:
- The first term is [tex]\(-35x^5\)[/tex]. Divide it by [tex]\(-5x\)[/tex]:
[tex]\[
\frac{-35x^5}{-5x} = 7x^{5-1} = 7x^4
\][/tex]
- The second term is [tex]\(-40x^4\)[/tex]. Divide it by [tex]\(-5x\)[/tex]:
[tex]\[
\frac{-40x^4}{-5x} = 8x^{4-1} = 8x^3
\][/tex]
- The third term is [tex]\(15x^3\)[/tex]. Divide it by [tex]\(-5x\)[/tex]:
[tex]\[
\frac{15x^3}{-5x} = -3x^{3-1} = -3x^2
\][/tex]
- The fourth term is [tex]\(-15x^2\)[/tex]. Divide it by [tex]\(-5x\)[/tex]:
[tex]\[
\frac{-15x^2}{-5x} = 3x^{2-1} = 3x
\][/tex]

3. Combine the Results:
Now, we combine all the individual results to obtain the quotient:
[tex]\[
7x^4 + 8x^3 - 3x^2 + 3x
\][/tex]

This polynomial, [tex]\(7x^4 + 8x^3 - 3x^2 + 3x\)[/tex], is the quotient of the original division problem. There is no remainder in this case, as the division is exact.

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Rewritten by : Barada