How do you divide #(4x^3-5x^2-4x-12)/(3x-4) #?

Question

How do you divide #(4x^3-5x^2-4x-12)/(3x-4)  #?

Julia Adams · Accepted Answer

#4/3 x^2 + 1/9x - 32/27 " Remainder: " (-196)/(27(3x-4))# This is best solved using long division. The question is, what do I have to multiply by #3x-4# to get #4x^3-5x^2-4x-12#? #" "4/3 x^2 + 1/9x - 32/27 " Remainder: " (-196)/(27(3x-4))# #3x-4 |bar(4x^3-5x^2-4x-12)# #" "-ul((4x^3-16/3 x^2)) downarrow " " downarrow# #" "1/3x^2-4x# #" "-ul((1/3x^2-4/9x))# #" "-32/9x-12# #" "-ul((-32/9x+128/27))# #" "-196/27#

Nathan Axel · Answer

#color(magenta)(1.3x^2+0.1x-1.2# and remainder of #color(magenta)(-6# #(4x^3-5x^2-4x-12)/(3x-4)# # color(white)(................)color(magenta)(1.3x^2+0.1x-1.2# #color(white)(a)3x-4##|##overline(4x^3-5x^2-4x-12)# #color(white)(..............)ul(4x^3-5.3x^2)# #color(white)(........................)0.3x^2-4x# #color(white)(........................)ul(0.3x^2-0.4x)# #color(white)(...............................)-3.6x-12# #color(white)(................................)ul(-3.6x+4.8)# #color(white)(...........................................)color(magenta)(-6# #color(magenta)((4x^3-5x^2-4x-12)/(3x-4)=1.3x^2+0.1x-1.2#and remainder of #color(magenta)(-6#

Ethan Adkins · Answer

undefined To divide (4x^3-5x^2-4x-12) by (3x-4), you can use long division. Here are the steps:

1. Divide the first term of the numerator (4x^3) by the first term of the denominator (3x). The result is (4/3)x^2.

2. Multiply the entire denominator (3x-4) by the result from step 1, which is (4/3)x^2. This gives you (4/3)x^2(3x-4) = (4/3)x^3 - (16/3)x^2.

3. Subtract the result from step 2 from the original numerator (4x^3-5x^2-4x-12). This gives you a new polynomial: (-5/3)x^2 - 4x - 12.

4. Repeat steps 1-3 with the new polynomial (-5/3)x^2 - 4x - 12.

5. Divide the first term of the new polynomial (-5/3)x^2 by the first term of the denominator (3x). The result is (-5/9)x.

6. Multiply the entire denominator (3x-4) by the result from step 5, which is (-5/9)x. This gives you (-5/9)x(3x-4) = (-5/3)x^2 + (20/9)x.

7. Subtract the result from step 6 from the new polynomial (-5/3)x^2 - 4x - 12. This gives you a new polynomial: (-4/9)x - 12.

8. Repeat steps 1-3 with the new polynomial (-4/9)x - 12.

9. Divide the first term of the new polynomial (-4/9)x by the first term of the denominator (3x). The result is (-4/27).

10. Multiply the entire denominator (3x-4) by the result from step 9, which is (-4/27). This gives you (-4/27)(3x-4) = (-4/9)x + (16/27).

11. Subtract the result from step 10 from the new polynomial (-4/9)x - 12. This gives you a remainder of (-16/27).

Therefore, the division of (4x^3-5x^2-4x-12) by (3x-4) is equal to (4/3)x^2 + (-5/9)x + (-4/27) with a remainder of (-16/27).