How do you divide #(7x^4 - 4x^2 – 36x+ 81 )/((x + 9) )#?

Question

How do you divide #(7x^4 - 4x^2 – 36x+ 81 )/((x  + 9) )#?

Isaiah Anthony · Accepted Answer

#7x^4-4x^2-36+81#

#= (x+9)(7x^3-63x^2+563x-5103)+46008#
I like to long divide the coefficients, not forgetting to include a #0# for any missing power of #x# (in this case #x^3#)...

Long division of coefficients is similar to long division of numbers.

We find:

#7x^4-4x^2-36+81#

#= (x+9)(7x^3-63x^2+563x-5103)+46008#
That is #(7x^4-4x^2-36+81)# divided by #(x+9)# results in a quotient #(7x^3-63x^2+563x-5103)# with remainder #46008#.

Julia Adams · Answer

undefined To divide (7x^4 - 4x^2 – 36x + 81) by (x + 9), you can use long division. Here are the steps:

1. Divide the first term of the numerator (7x^4) by the first term of the denominator (x). This gives you 7x^3.
2. Multiply the entire denominator (x + 9) by the result from step 1 (7x^3), and write the product (7x^3(x + 9)) below the numerator.
3. Subtract the product from the numerator: (7x^4 - 4x^2 – 36x + 81) - (7x^3(x + 9)).
   Simplify this expression: 7x^4 - 4x^2 – 36x + 81 - 7x^3(x + 9).
4. Bring down the next term from the numerator, which is -4x^2.
5. Divide the new expression (-4x^2 – 36x + 81) by the first term of the denominator (x). This gives you -4x.
6. Multiply the entire denominator (x + 9) by the result from step 5 (-4x), and write the product (-4x(x + 9)) below the expression from step 5.
7. Subtract the product from the expression: (-4x^2 – 36x + 81) - (-4x(x + 9)).
   Simplify this expression: -4x^2 – 36x + 81 + 4x(x + 9).
8. Bring down the next term from the numerator, which is -36x.
9. Repeat steps 5-7 with the new expression (-36x + 81).
10. Bring down the last term from the numerator, which is 81.
11. Repeat steps 5-7 with the new expression (81).
12. At this point, you have no more terms to bring down. The division is complete.
13. The quotient is the sum of the results from each step: 7x^3 - 4x - 36 + (81/(x + 9)).

Therefore, the division of (7x^4 - 4x^2 – 36x + 81) by (x + 9) is equal to 7x^3 - 4x - 36 + (81/(x + 9)).