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

Question

Abigail Allen · Accepted Answer

#4x^2 - 24x + 77, R -228#

polynomial long division - yay!

Avery Alexander · Answer

#4x^2-24x+77-228/(x+3)# #"one way is to use the divisor as a factor in the numerator"# #"consider the numerator"# #color(red)(4x^2)(x+3)color(magenta)(-12x^2)-12x^2+5x+3# #=color(red)(4x^2)(x+3)color(red)(-24x)(x+3)color(magenta)(+72x)+5x+3# #=color(red)(4x^2)(x+3)color(red)(-24x)(x+3)color(red)(+77)(x+3)color(magenta)(-231)+3# #=color(red)(4x^2)(x+3)color(red)(-24x)(x+3)color(red)(+77)(x+3)-228# #"quotient "=color(red)(4x^2-24x+77)," remainder "=-228# #rArr(4x^3-12x^2+5x+3)/(x+3)# #=4x^2-24x+77-228/(x+3)#

Eli Assouline · Answer

undefined To divide (4x^3 - 12x^2 + 5x + 3) by (x + 3), you can use long division or synthetic division. Here is the step-by-step process using long division:

1. Divide the first term of the dividend (4x^3) by the first term of the divisor (x). This gives you 4x^2.
2. Multiply the divisor (x + 3) by the quotient obtained in step 1 (4x^2). This gives you 4x^3 + 12x^2.
3. Subtract the result obtained in step 2 from the dividend (4x^3 - 12x^2 + 5x + 3) to get a new polynomial: (-12x^2 + 5x + 3) - (4x^3 + 12x^2) = -4x^3 - 24x^2 + 5x + 3.
4. Bring down the next term from the dividend, which is 5x.
5. Divide the new polynomial (-4x^3 - 24x^2 + 5x + 3) by the first term of the divisor (x). This gives you -4x^2.
6. Multiply the divisor (x + 3) by the quotient obtained in step 5 (-4x^2). This gives you -4x^3 - 12x^2.
7. Subtract the result obtained in step 6 from the new polynomial (-4x^3 - 24x^2 + 5x + 3) to get a new polynomial: (-24x^2 + 5x + 3) - (-4x^3 - 12x^2) = 4x^3 - 12x^2 + 5x + 3.
8. Bring down the next term from the dividend, which is 3.
9. Divide the new polynomial (4x^3 - 12x^2 + 5x + 3) by the first term of the divisor (x). This gives you 4x.
10. Multiply the divisor (x + 3) by the quotient obtained in step 9 (4x). This gives you 4x^2 + 12x.
11. Subtract the result obtained in step 10 from the new polynomial (4x^3 - 12x^2 + 5x + 3) to get a new polynomial: (-12x^2 + 5x + 3) - (4x^2 + 12x) = -16x^2 - 7x + 3.

Therefore, the quotient is 4x^2 - 4x + 1, and the remainder is -16x^2 - 7x + 3.