Building Sentences: Dividing Polynomials

Dividing polynomials involves long division. We determine what we need to multiply the first term in the divisor by to get the first term in the dividend. Then multiply the entire divisor and subtract from the dividend. We line up powers of x in the quotient just like we lined up powers of 10 when we did long division of real numbers.

x+1

)

x3y+

x2y+

x2+

xy+

2x+

y+

1

Now I ask what do I need to multiply x by to get x³y?. The answer is x²y. I multiply x+1 by x²y and subtract.

			x2y
x+1	)	x3y+	x2y+	x2+	xy+	2x+	y+	1
		-x3y	-x2y

Now I bring down the rest of the dividend and ask what do I multiply x by to get x²? The answer is x. I multiply x+1 by x and subtract.

			x2y+	x
x+1	)	x3y+	x2y+	x2+	xy+	2x+	y+	1
		-x3y	-x2y
				x2+	xy+	2x+	y+	1
				-x2		-x

Now I bring down the rest of the dividend and ask what do I multiply x by to get xy? The answer is y. I multiply x+1 by y and subtract.

			x2y+	x+			y
x+1	)	x3y+	x2y+	x2+	xy+	2x+	y+	1
		-x3y	-x2y
				x2+	xy+	2x+	y+	1
				-x2		-x
					xy+	x+	y+	1
					-xy		-y

When I bring down what's left of the dividend I get x+1. If I multiply x+1 by 1 I get x+1. So I multiply x+1 by 1 and subtract. This time it comes out even.

			x2y+	x+			y+	1
x+1	)	x3y+	x2y+	x2+	xy+	2x+	y+	1
		-x3y	-x2y
				x2+	xy+	2x+	y+	1
				-x2		-x
					xy+	x+	y+	1
					-xy		-y
						x+		1
						-x		-1

So, (x³y+x²y+x²+xy+2x+y+1)÷(x+1)=x²y+x+y+1.