How do you find the quotient of #(x^3+27) / (x+3)#?

Question

Abigail Allen · Accepted Answer

#x^2-3x+9#

Long division of polynomials can be used. First, arrange the problem in long division format, with the numerator under the division sign and the denominator in front.

#color(white)(x/color(black)(x+3)) ()/(")" x^3 +0x^2 + 0x + 27)#

To start the division, look at the first term of each term. What do you need to multiply #x# by to get #x^3#? The answer is #x^2#, so that goes on the top.

#color(white)(x/color(black)(x+3)) (x^2color(white)(-3x -9 +27x))/(")" x^3 +0x^2 + 0x + 27)#

Now multiply the divisor, #x+3#, by #x^2# and subtract from the dividend. Remember, subtraction happens in columns.

#color(white)(x/color(black)(x+3)) (x^2 color(white)(-3x -9 +27x))/(")" x^3 +0x^2 + 0x + 27)# #color(white)(XXXx)(x^3 + 3x^2)/(color(white)(x^3)-3x^2)#

Now we need to know what to multiply #x# by to get #-3x^2#. Its #-3x#. Write #-3x# on top, multiply by #x+3# and subtract.

#color(white)(x/color(black)(x+3)) (x^2-3x color(white)(+9 +27x))/(")" x^3 +0x^2 + 0x + 27)# #color(white)(XXXx)(x^3 + 3x^2)/(color(white)(x^3)-3x^2)# #color(white)(XXXXX|)(-3x^2-9x)/(color(white)(-3x^2-)9x#

Last one, we multiply #x# by #9# to get #9x#.

#color(white)(x/color(black)(x+3)) (x^2-3x +9 color(white)(+27x))/(")" x^3 +0x^2 + 0x + 27)# #color(white)(XXXx)(x^3 + 3x^2)/(color(white)(x^3)-3x^2)# #color(white)(XXXXX|)(-3x^2-9x)/(color(white)(-3x^2-)9x# #color(white)(XXXXXXXXXx)(9x+27)/(color(white)(9x+)0)#

The quotient is since there is no remainder;

#x^2-3x+9#

Julia Adams · Answer

undefined To find the quotient of (x^3+27) / (x+3), we can use polynomial long division or synthetic division. Let's use polynomial long division:

Step 1: Divide x^3 by x, which gives us x^2.
Step 2: Multiply x+3 by x^2, which gives us x^3+3x^2.
Step 3: Subtract (x^3+3x^2) from (x^3+27), which gives us -3x^2+27.
Step 4: Bring down the next term, which is 0.
Step 5: Divide -3x^2 by x, which gives us -3x.
Step 6: Multiply x+3 by -3x, which gives us -3x^2-9x.
Step 7: Subtract (-3x^2-9x) from (-3x^2+27), which gives us 9x.
Step 8: Bring down the next term, which is 0.
Step 9: Divide 9x by x, which gives us 9.
Step 10: Multiply x+3 by 9, which gives us 9x+27.
Step 11: Subtract (9x+27) from (9x+0), which gives us -27.

Therefore, the quotient of (x^3+27) / (x+3) is x^2 - 3x + 9 with a remainder of -27.