Answer :
To find the quotient of the polynomial [tex]\(-35x^5 - 40x^4 + 15x^3 - 15x^2\)[/tex] divided by [tex]\(-5x^2\)[/tex], follow these steps:
1. Divide Each Term Individually: You can divide each term of the polynomial by the divisor [tex]\(-5x^2\)[/tex].
- For the first term [tex]\(-35x^5\)[/tex]:
[tex]\[
\frac{-35x^5}{-5x^2} = 7x^{5-2} = 7x^3
\][/tex]
- For the second term [tex]\(-40x^4\)[/tex]:
[tex]\[
\frac{-40x^4}{-5x^2} = 8x^{4-2} = 8x^2
\][/tex]
- For the third term [tex]\(15x^3\)[/tex]:
[tex]\[
\frac{15x^3}{-5x^2} = -3x^{3-2} = -3x
\][/tex]
- For the fourth term [tex]\(-15x^2\)[/tex]:
[tex]\[
\frac{-15x^2}{-5x^2} = 3x^{2-2} = 3
\][/tex]
2. Write Out the Quotient: Combine all the terms you've calculated.
- Putting it all together, the quotient is:
[tex]\[
7x^3 + 8x^2 - 3x + 3
\][/tex]
This is the simplified quotient of the division of the polynomial [tex]\(-35x^5 - 40x^4 + 15x^3 - 15x^2\)[/tex] by [tex]\(-5x^2\)[/tex].
1. Divide Each Term Individually: You can divide each term of the polynomial by the divisor [tex]\(-5x^2\)[/tex].
- For the first term [tex]\(-35x^5\)[/tex]:
[tex]\[
\frac{-35x^5}{-5x^2} = 7x^{5-2} = 7x^3
\][/tex]
- For the second term [tex]\(-40x^4\)[/tex]:
[tex]\[
\frac{-40x^4}{-5x^2} = 8x^{4-2} = 8x^2
\][/tex]
- For the third term [tex]\(15x^3\)[/tex]:
[tex]\[
\frac{15x^3}{-5x^2} = -3x^{3-2} = -3x
\][/tex]
- For the fourth term [tex]\(-15x^2\)[/tex]:
[tex]\[
\frac{-15x^2}{-5x^2} = 3x^{2-2} = 3
\][/tex]
2. Write Out the Quotient: Combine all the terms you've calculated.
- Putting it all together, the quotient is:
[tex]\[
7x^3 + 8x^2 - 3x + 3
\][/tex]
This is the simplified quotient of the division of the polynomial [tex]\(-35x^5 - 40x^4 + 15x^3 - 15x^2\)[/tex] by [tex]\(-5x^2\)[/tex].
