How do you simplify #(2x^3-14x+2 )/(x+3)#?

Question

How do you simplify #(2x^3-14x+2  )/(x+3)#?

Genesis Allen · Accepted Answer

I agree with Jim that the expression maybe should have been

#(2x^3-14x+12) / (x+3)#,

but let's see what happens with the one given...

#(2x^3-14x+2) / (x+3)# #=(2x^3+6x^2-6x^2-18x+4x+12-10) / (x+3)# #=((2x^2-6x+4)(x+3)-10) / (x+3)# #=2x^2-6x+4 - 10/(x+3)# #=2(x-2)(x-1) - 10/(x+3)# This doesn't really tell you much more than the original expression, and is not noticeably simpler. It would be helpful if that #-10/(x+3)# remainder term were not there, which is what would happen if the numerator were #2x^3-14x+12#

Ethan Adkins · Answer

undefined To simplify the expression (2x^3-14x+2)/(x+3), you can use polynomial long division or synthetic division. However, I will demonstrate the process using polynomial long division:

Step 1: Divide the first term of the numerator (2x^3) by the first term of the denominator (x). This gives you 2x^2.

Step 2: Multiply the entire denominator (x+3) by the quotient obtained in the previous step (2x^2). This gives you 2x^2(x+3) = 2x^3 + 6x^2.

Step 3: Subtract the result obtained in step 2 from the numerator (2x^3-14x+2) to get the remainder. This gives you (-14x+2) - (2x^3 + 6x^2) = -2x^3 - 6x^2 - 14x + 2.

Step 4: Bring down the next term from the numerator (-2x^3) and repeat steps 1-3.

Step 5: Divide the first term of the new numerator (-2x^3) by the first term of the denominator (x). This gives you -2x^2.

Step 6: Multiply the entire denominator (x+3) by the new quotient obtained in step 5 (-2x^2). This gives you -2x^2(x+3) = -2x^3 - 6x^2.

Step 7: Subtract the result obtained in step 6 from the new numerator (-2x^3 - 6x^2 - 14x + 2) to get the new remainder. This gives you (-14x+2) - (-2x^3 - 6x^2) = -2x^3 + 6x^2 - 14x + 2.

Step 8: Repeat steps 4-7 until there are no more terms left in the numerator.

The simplified form of the expression (2x^3-14x+2)/(x+3) is 2x^2 - 2x + 6 - (2x^3 + 6x^2 - 14x + 2)/(x+3).