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

Question

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

Eva Adamson · Accepted Answer

The result is #(-x^2-3x+1)# remainder #5x-10#.

Looking at the orders of the numerator and the denominator, we can see the result should be of order 2. We can deduce the first term will be #-x^2# in order for the first term of the product to be #-x^4#. Looking at the constants in the numerator it seems there must be a remainder, since 3 is not a factor of 7. We write the numerator as a product of two quadratics (the known denominator and the unknown quotient) plus a linear term (the unknown remainder): #(-x^4-3x^3-2x^2-4x-7)# #=(x^2+3)(-x^2+Ax+B)+Cx+D# Comparing coefficients of #x^3#, we see that: #A=-3# Comparing coefficients of #x^2#, we see that: #B-3=-2 rArr B=1# Comparing coefficients of #x# ,we see that: #3A+C=-4 rArr C=5# Comparing constants, we see that: #3B+D=-7 rArr D=-10# Hence the result is #(-x^2-3x+1)# remainder #5x-10#.

Nathan Axel · Answer

#-x^2-3x+1+(5x-10)/(x^2+3)#

#(-x^4-3x^3-2x^2-4x-7 )/ (color(green)(x^2+3))#

Note that I use place keepers such as #0x^3# for eas of alignment.

#color(white)("dddddddddddddd") -x^4-3x^3-2x^2-4x-7 # #color(magenta)(-x^2) color(green)((x^2+3)) ->color(white)("d") ul(-x^4+0x^3-3x^2larr" Subtract")# # color(white)("dddddddddddddddd")0-3x^3 +color(white)("d")x^2-4x-7# #color(magenta)(-3x)color(green)((x^2+3))->color(white)("ddd.d") ul(-3x^3+0x^2-9xlarr" Subtract")# #color(white)("dddddddddddddddddddd")0color(white)("d")+color(white)("d")x^2+5x-7# #color(white)(1)color(magenta)(+1)color(green)((x^2+3)) ->color(white)("ddddddddddd.d")ul( x^2+0x+3larr" Subtract")# #color(white)("dddddddddddddddddddddddddd")0+color(magenta)(5x-10)larr" Stop"#

Combining everything, we get:

#color(magenta)(-x^2-3x+1+(5x-10)/(color(black)(color(green)(x^2+3)))#

Jameson Allen · Answer

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

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

2. Multiply the entire denominator (x^2+3) by -x^2, and subtract the result from the numerator. This gives you a new numerator: (-x^4-3x^3-2x^2-4x-7) - (-x^2)(x^2+3).

3. Simplify the new numerator: -3x^3-2x^2-4x-7 + (x^4+3x^2).

4. Repeat steps 1-3 with the new numerator. Divide the first term of the new numerator (-3x^3) by the first term of the denominator (x^2). The result is -3x.

5. Multiply the entire denominator (x^2+3) by -3x, and subtract the result from the new numerator. This gives you a new numerator: -3x^3-2x^2-4x-7 - (-3x)(x^2+3).

6. Simplify the new numerator: -2x^2-4x-7 + (3x^3+9x).

7. Repeat steps 1-3 with the new numerator. Divide the first term of the new numerator (-2x^2) by the first term of the denominator (x^2). The result is -2.

8. Multiply the entire denominator (x^2+3) by -2, and subtract the result from the new numerator. This gives you a new numerator: -2x^2-4x-7 - (-2)(x^2+3).

9. Simplify the new numerator: -4x-7 + (2x^2+6).

10. Repeat steps 1-3 with the new numerator. Divide the first term of the new numerator (-4x) by the first term of the denominator (x^2). The result is -4x.

11. Multiply the entire denominator (x^2+3) by -4x, and subtract the result from the new numerator. This gives you a new numerator: -4x-7 - (-4x)(x^2+3).

12. Simplify the new numerator: -7 + (4x^3+12x).

13. Repeat steps 1-3 with the new numerator. Divide the first term of the new numerator (-7) by the first term of the denominator (x^2). The result is 0.

14. Multiply the entire denominator (x^2+3) by 0, and subtract the result from the new numerator. This gives you a new numerator: -7 - 0(x^2+3).

15. Simplify the new numerator: -7.

The final result of the division is -x^2 - 3x - 2 + (-7)/(x^2+3).