How do you solve this problem Divide polynomial

Question

When you divide x^4−6x^3+12x^2−21x+20 by x-4 the quotient is x^3−2x^2+_[blank]_ -5. What is the missing term in the quotient?

Answer 1

When dividing by a polynomial you are concerned with the variable first. In this problem we have

x⁴-6x³+12x²-21x+20 divided by x-4. If you write this out in long division form it will be easier to follow:

x³-2x²+4x-5

x-4Γx⁴-6x³+12x²-21x+20

-(x⁴-4x³) <----subtract product of (x-4) times x³

-2x³+12x² <----result is (-2x³) then bring down the 12x²

-(-2x³+8x²) <----subtract product of (x-4) times (-2x²)

4x²-21x <----result is 4x² then bring down the (-21x)

-(4x²-16x) <---subtract product of (x-4) times 4x

-5x+20 <---result is (-5x) then bring down the 20

-(5x+20) <---subtract product of (x-4) times (-5)

0 <---result is "0" which means there is no remainder

Therefore, the answer quotient is x³-2x²+4x-5 and the answer you are looking for is 4x.

Hope this helps!