Long division polynomials?

Question

Using long division can you please walk me through these problems step by step:
a). (x^3-7x-6) / (x+1)

b). (x^2+4x+4) / (x+2)

thanks in advance.

Answer 1

Long division of polynomials works just like long division of numbers.

a) you are dividing x^3 - 7x - 6 by x+1

x^3 is the highest exponent of x term in the numerator, x is the highest exponent of x term in the denominator, so your first part of the quotient will be x^3/x = x^2.

Multiply that by the denominator:

x^2(x+1) = x^3 + x^2

Subtract that from the original equation:

x^3 - 7x - 6 - (x^3 + x^2) = -x^2 -7x - 6

-x^2 is the highest exponent of x term in the numerator, x is the highest exponent of x term in the denominator, so your second part of the quotient will be -x^2/x = -x.

Multiply that by the denominator:

(-x) * (x+1) = -x^2 - x

Subtract that from the remainder in the previous step:

-x^2 -7x - 6 - (-x^2 - x) = -6x - 6

This is just -6 times (x+1), so the final part of the quotient is -6, and the remainder is 0.

Your answer is x^2 - x - 6, which are the three values calculated in the three steps above.

b) you are dividing x^2+4x+4 by x+2

x^2 is the highest exponent of x term in the numerator, x is the highest exponent of x term in the denominator, so your first part of the quotient will be x^2/x = x.

Multiply that by the denominator:

x * (x+2) = x^2 + 2x

Subtract that from the original equation:

x^2 + 4x + 4 - (x^2 + 2x) = 2x + 4

This is just 2 times (x+2), so the final part of the quotient is 2, and the remainder is 0.

Your answer is x + 2, which are the two values calculated in the two steps above.

Answer 2

You could do this with factoring, but factoring uses our brains so there is this process called synthetic division to shorten it down. You first put the coefficients of each value and divide by the term x-r
okay so it turns out to be like this x+1=x-(-1) so -1 = r. The coefficients of the polynomial turns into,
1 for x^3, 0 for x^2, -7 for 7x, and eventually -6 for -6.
You set up the division like this :
-1| 1 0 -7 -6
okay you pull the first value down.
-1| 1 0 -7 -6
0
1
Then you multiply your (r) by the number brought down and add it to your next number
-1| 1 0 -7 -6
0 -1
1 -1
You follow the step
-1| 1 0 -7 -6
0 -1 1 6
1 -1 -6 0
the powers of all you numbers have dropped by one so your answer is x^2 - x - 6 the 0 is your remainder so it would have been 0/x+1 but thats useless. This also meant your first divisor is a factor of the polynomial.
b) x^2 + 4x+ 4/ x+2
Set up your numbers again. x-(-2), -2 = r
1 for the coefficient of x^2, 4 for 4x and 4 for 4.
-2| 1 4 4
0 -2 -4
1 2 0
and you finally get x+ 2x. * subtract power by 1 unless you aren't dividing by x-r and rather x^2-r you would have to convert a lot
I could show you the factoring method for this one because its easier. (x+2)(x+2)= x^2+4x+4 so (x+2)(x+2)/X=2= x+2

Answer 3

x^2 -3xy + 2y^2 + 3x - 6y - 8 | x - y + 4, x (coef for x first) - x^2 + xy - 4x ====================== -2xy + 2y^2 - x - 6y - 8 (reorder now) 2y^2 - 2xy - x - 6y -8 | x - y + 4, -2y (coef for y 2nd) 2xy - 2y^2 + 8y ===================== -x + 2y -8 | x - y + 4, -a million x - y + 4 ============= y - 4 <- the rest quotient = x - 2y -a million ..... = x - 2y -a million + (y - 4)/(x - y + 4) are you particular for the coef of x? is 3? or 2? regardless of if that's 2 you will get a clean result :) ... = x - 2y - 2

Answer 4

a)(x^3-7x-6)/(x+1)

_____x^2-x-6_
x+1|x^3+0x^2-7x-6
-x^3-x^2
======
-x^2-7x
+x^2+x
======
-6x-6
+6x+6
=====
0

The answer is x^2-x-6.

b)(x^2+4x+4)/(x+2)

____x+2_
x+2|x^2+4x+4
-x^2-2x
======
2x+4
-2x-4
====
0

The answer is (x+2).