Polynomial long division

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

Explaining in more details the polynomial long division in the figure above

The polynomial long division above is a summary showing how to do long division with polynomials. Read example #1 to fully understand it!

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

Another straightforward example showing polynomial long division

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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