How do you divide #(x^4 - 7x^3 + 2x^2 + 9x)/(x^3-x^2+2x+1)#?

Question

Nathan Axel · Accepted Answer

Long divide the coefficients to find:

#(x^4-7x^3+2x^2+9x)/(x^3-x^2+2x+1) = x-6 + (-6x^2+4x+6)/(x^3-x^2+2x+1)#

That is, the quotient is #x-6# with remainder #-6x^2+4x+6#

I like to long divide the coefficients, not forgetting to include #0#'s for any missing powers of #x#. In this example, that means the constant term of the dividend...

The process is similar to long division of numbers.

Write the dividend (#1, -7, 2, 9, 0#) under the bar and the divisor (#1, -1, 2, 1#) to the left.

Choose the first term #color(blue)(1)# of the quotient so that when multiplied by the divisor it matches the leading term of the dividend.

Write out the product of this first term of the quotient and the divisor below the dividend and subtract it.

Bring down the next term (#0#) of the dividend alongside it.

Choose the next term #color(blue)(-6)# of the quotient so that when multiplied by the divisor it matches the leading term of the remainder.

Write out the product of this second term of the quotient and the divisor below the remainder and subtract it.

The resulting remainder #-6, 4, 6# is shorter than the divisor and there are no more terms to bring down from the dividend. So this is our final remainder.

In this example, we find the quotient is #x-6# and remainder #-6x^2+4x+6#

Kennedy Adams · Answer

undefined To divide (x^4 - 7x^3 + 2x^2 + 9x) by (x^3 - x^2 + 2x + 1), you can use long division. Here are the steps:

1. Divide the highest degree term of the numerator (x^4) by the highest degree term of the denominator (x^3). The result is x.

2. Multiply the entire denominator (x^3 - x^2 + 2x + 1) by the result from step 1 (x). This gives you x^4 - x^3 + 2x^2 + x.

3. Subtract the result from step 2 from the numerator (x^4 - 7x^3 + 2x^2 + 9x). This gives you -6x^3 + 9x.

4. Bring down the next term from the numerator (-6x^3 + 9x). This gives you -6x^3 + 9x + 0.

5. Divide the highest degree term of the new numerator (-6x^3) by the highest degree term of the denominator (x^3). The result is -6x^2.

6. Multiply the entire denominator (x^3 - x^2 + 2x + 1) by the result from step 5 (-6x^2). This gives you -6x^3 + 6x^2 - 12x - 6.

7. Subtract the result from step 6 from the new numerator (-6x^3 + 9x + 0). This gives you 15x^2 - 3x - 6.

8. Bring down the next term from the numerator (15x^2 - 3x - 6). This gives you 15x^2 - 3x - 6 + 0.

9. Divide the highest degree term of the new numerator (15x^2) by the highest degree term of the denominator (x^3). The result is 0.

10. Multiply the entire denominator (x^3 - x^2 + 2x + 1) by the result from step 9 (0). This gives you 0.

11. Subtract the result from step 10 from the new numerator (15x^2 - 3x - 6 + 0). This gives you 15x^2 - 3x - 6.

Since the degree of the new numerator (15x^2 - 3x - 6) is less than the degree of the denominator (x^3 - x^2 + 2x + 1), the division is complete. The quotient is x - 6x^2 and the remainder is 15x^2 - 3x - 6.