# Polynomial long division

Learn polynomial long division with these two examples that are easy to follow and straight to the point.

Example #1

Divide x2 + 3x - 10 by x - 2

Divide the leading term of x2 + 3x - 10 by the leading term of x - 2

x2 ÷ x = x

Write x as the first term of the quotient

x
x - 2)  x2 + 3x - 10

Multiply the first term of the quotient by the divisor

x(x - 2) = x2 - 2x

Subtract x2 - 2x from the dividend

x
x - 2)  x2 + 3x - 10
-( x2 - 2x)

x
x - 2)  x2 + 3x - 10
-x2 + 2x
______________
5x

Bring down -10

x
x - 2)  x2 + 3x - 10
-x2 + 2x
______________
5x  - 10

Divide the leading term of 5x - 10 by the leading term of x - 2

5x ÷ x = 5

Write 5 as the second term of the quotient. Since 5 is positive, you can put a + sign between the first term and the second term.

x  + 5
x - 2)  x2 + 3x - 10
-x2 + 2x
______________
5x  - 10

Multiply the second term of the quotient by the divisor

5(x - 2) = 5x - 10

Subtract 5x - 10 from 5x - 10

x  + 5
x - 2)  x2 + 3x - 10
-x2 + 2x
______________
5x  - 10
-5x + 10
__________
0

(x2 + 3x - 10) ÷ (x - 2) = x + 5

Example #2

Divide x2 - 5x + 1 by x + 3

Divide the leading term of x2 - 5x + 1 by the leading term of x + 3

x2 ÷ x = x

Write x as the first term of the quotient

x
x + 3)  x2 - 5x + 1

Multiply the first term of the quotient by the divisor

x(x + 3) = x2 + 3x

Subtract x2 + 3x from the dividend

x
x + 3)  x2 - 5x + 1
-( x2 + 3x)

x
x + 3)  x2 - 5x + 1
-x2 - 3x
______________
-8x

Bring down 1

x
x + 3)  x2 - 5x + 1
-x2 - 3x
______________
-8x  + 1

Divide the leading term of -8x  + 1 by the leading term of x + 3

-8x ÷ x = -8

Write -8 as the second term of the quotient. Notice that putting a + sign between the first term and the second term does not change the problem.

x + -8
x + 3)  x2 - 5x + 1
-x2 - 3x
______________
-8x  + 1

Multiply the second term of the quotient by the divisor

-8(x + 3) = -8x - 24

Subtract -8x - 24 from -8x + 1

x  + -8
x + 3)  x2 - 5x + 1
-x2 - 3x
______________
-8x  + 1
8x + 24
__________
25

(x2 - 5x + 1) ÷ (x + 3) = x + -8 with a remainder of 25

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