Answer :
To divide the polynomial [tex]\(8x^7 - 9x^6 + 20x^5 - 4x^4\)[/tex] by the monomial [tex]\(-4x^4\)[/tex], you can divide each term in the polynomial individually by the monomial. Here's how you do it, step by step:
1. Divide each term:
- First term: [tex]\(\frac{8x^7}{-4x^4}\)[/tex]
- Divide the coefficients: [tex]\(\frac{8}{-4} = -2\)[/tex]
- Subtract the exponents of like bases: [tex]\(x^{7-4} = x^3\)[/tex]
- Result: [tex]\(-2x^3\)[/tex]
- Second term: [tex]\(\frac{-9x^6}{-4x^4}\)[/tex]
- Divide the coefficients: [tex]\(\frac{-9}{-4} = \frac{9}{4}\)[/tex]
- Subtract the exponents: [tex]\(x^{6-4} = x^2\)[/tex]
- Result: [tex]\(\frac{9}{4}x^2\)[/tex]
- Third term: [tex]\(\frac{20x^5}{-4x^4}\)[/tex]
- Divide the coefficients: [tex]\(\frac{20}{-4} = -5\)[/tex]
- Subtract the exponents: [tex]\(x^{5-4} = x^1\)[/tex] or [tex]\(x\)[/tex]
- Result: [tex]\(-5x\)[/tex]
- Fourth term: [tex]\(\frac{-4x^4}{-4x^4}\)[/tex]
- Divide the coefficients: [tex]\(\frac{-4}{-4} = 1\)[/tex]
- The exponents are the same, so [tex]\(x^{4-4} = x^0 = 1\)[/tex]
- Result: [tex]\(1\)[/tex]
2. Combine the results:
- The complete result of dividing the polynomial by the monomial is:
[tex]\(-2x^3 + \frac{9}{4}x^2 - 5x + 1\)[/tex]
3. Check the division:
To ensure the division is correct, multiply each term of the result by the monomial [tex]\(-4x^4\)[/tex] and add them together. If the original polynomial is obtained, the division was done correctly:
- [tex]\((-2x^3)(-4x^4) = 8x^7\)[/tex]
- [tex]\(\left(\frac{9}{4}x^2\right)(-4x^4) = -9x^6\)[/tex]
- [tex]\((-5x)(-4x^4) = 20x^5\)[/tex]
- [tex]\(1(-4x^4) = -4x^4\)[/tex]
Add them up: [tex]\(8x^7 - 9x^6 + 20x^5 - 4x^4\)[/tex], which matches the original polynomial. Therefore, the division is correct.
1. Divide each term:
- First term: [tex]\(\frac{8x^7}{-4x^4}\)[/tex]
- Divide the coefficients: [tex]\(\frac{8}{-4} = -2\)[/tex]
- Subtract the exponents of like bases: [tex]\(x^{7-4} = x^3\)[/tex]
- Result: [tex]\(-2x^3\)[/tex]
- Second term: [tex]\(\frac{-9x^6}{-4x^4}\)[/tex]
- Divide the coefficients: [tex]\(\frac{-9}{-4} = \frac{9}{4}\)[/tex]
- Subtract the exponents: [tex]\(x^{6-4} = x^2\)[/tex]
- Result: [tex]\(\frac{9}{4}x^2\)[/tex]
- Third term: [tex]\(\frac{20x^5}{-4x^4}\)[/tex]
- Divide the coefficients: [tex]\(\frac{20}{-4} = -5\)[/tex]
- Subtract the exponents: [tex]\(x^{5-4} = x^1\)[/tex] or [tex]\(x\)[/tex]
- Result: [tex]\(-5x\)[/tex]
- Fourth term: [tex]\(\frac{-4x^4}{-4x^4}\)[/tex]
- Divide the coefficients: [tex]\(\frac{-4}{-4} = 1\)[/tex]
- The exponents are the same, so [tex]\(x^{4-4} = x^0 = 1\)[/tex]
- Result: [tex]\(1\)[/tex]
2. Combine the results:
- The complete result of dividing the polynomial by the monomial is:
[tex]\(-2x^3 + \frac{9}{4}x^2 - 5x + 1\)[/tex]
3. Check the division:
To ensure the division is correct, multiply each term of the result by the monomial [tex]\(-4x^4\)[/tex] and add them together. If the original polynomial is obtained, the division was done correctly:
- [tex]\((-2x^3)(-4x^4) = 8x^7\)[/tex]
- [tex]\(\left(\frac{9}{4}x^2\right)(-4x^4) = -9x^6\)[/tex]
- [tex]\((-5x)(-4x^4) = 20x^5\)[/tex]
- [tex]\(1(-4x^4) = -4x^4\)[/tex]
Add them up: [tex]\(8x^7 - 9x^6 + 20x^5 - 4x^4\)[/tex], which matches the original polynomial. Therefore, the division is correct.
