Answer :
To find the quotient of the given polynomial division, we're dividing [tex]\(-35x^5 - 40x^4 + 15x^3 - 15x^2\)[/tex] by [tex]\(-5x^2\)[/tex]. Here's how you can do it step-by-step:
1. Divide the first term of the numerator by the first term of the denominator:
[tex]\[
\frac{-35x^5}{-5x^2} = 7x^{3}
\][/tex]
This is the first term of the quotient.
2. Multiply the entire divisor by this new term and subtract from the original polynomial:
[tex]\[
(-5x^2) \times 7x^3 = -35x^5
\][/tex]
Subtract this from the original polynomial:
[tex]\[
(-35x^5 - 40x^4 + 15x^3 - 15x^2) - (-35x^5) = -40x^4 + 15x^3 - 15x^2
\][/tex]
3. Repeat the process with the new polynomial:
Now divide the new first term of the polynomial by the first term of the divisor:
[tex]\[
\frac{-40x^4}{-5x^2} = 8x^2
\][/tex]
This is the second term of the quotient.
4. Multiply the divisor by this new term and subtract:
[tex]\[
(-5x^2) \times 8x^2 = -40x^4
\][/tex]
Subtract this from the polynomial:
[tex]\[
(-40x^4 + 15x^3 - 15x^2) - (-40x^4) = 15x^3 - 15x^2
\][/tex]
5. Repeat the process again:
Divide the new first term by the first term of the divisor:
[tex]\[
\frac{15x^3}{-5x^2} = -3x
\][/tex]
This is the third term of the quotient.
6. Multiply the divisor by this new term and subtract:
[tex]\[
(-5x^2) \times -3x = 15x^3
\][/tex]
Subtract this from the polynomial:
[tex]\[
(15x^3 - 15x^2) - (15x^3) = -15x^2
\][/tex]
Once you reach a remainder polynomial of lower degree than the divisor, the division is complete.
The final quotient from the division is [tex]\(7x^3 + 8x^2 - 3x\)[/tex].
1. Divide the first term of the numerator by the first term of the denominator:
[tex]\[
\frac{-35x^5}{-5x^2} = 7x^{3}
\][/tex]
This is the first term of the quotient.
2. Multiply the entire divisor by this new term and subtract from the original polynomial:
[tex]\[
(-5x^2) \times 7x^3 = -35x^5
\][/tex]
Subtract this from the original polynomial:
[tex]\[
(-35x^5 - 40x^4 + 15x^3 - 15x^2) - (-35x^5) = -40x^4 + 15x^3 - 15x^2
\][/tex]
3. Repeat the process with the new polynomial:
Now divide the new first term of the polynomial by the first term of the divisor:
[tex]\[
\frac{-40x^4}{-5x^2} = 8x^2
\][/tex]
This is the second term of the quotient.
4. Multiply the divisor by this new term and subtract:
[tex]\[
(-5x^2) \times 8x^2 = -40x^4
\][/tex]
Subtract this from the polynomial:
[tex]\[
(-40x^4 + 15x^3 - 15x^2) - (-40x^4) = 15x^3 - 15x^2
\][/tex]
5. Repeat the process again:
Divide the new first term by the first term of the divisor:
[tex]\[
\frac{15x^3}{-5x^2} = -3x
\][/tex]
This is the third term of the quotient.
6. Multiply the divisor by this new term and subtract:
[tex]\[
(-5x^2) \times -3x = 15x^3
\][/tex]
Subtract this from the polynomial:
[tex]\[
(15x^3 - 15x^2) - (15x^3) = -15x^2
\][/tex]
Once you reach a remainder polynomial of lower degree than the divisor, the division is complete.
The final quotient from the division is [tex]\(7x^3 + 8x^2 - 3x\)[/tex].
