H Algebra 2/Trig · Unit 3
Divide the leading terms, multiply, and subtract. Keep going until the remainder is smaller than the divisor.
Jump to section:
Long division
Divide the leading terms. Multiply that result by the divisor. Subtract.
Repeat until the remainder has a smaller degree than the divisor.
The answer
Write quotient + remainder over the divisor. If the remainder is 0, the quotient is the whole answer.
Divide by x − c. The remainder is 0.
Divide x³ + 2x² − 4x + 1 by x − 1.
Divide the leading terms.
x³ ÷ x = x²
Multiply x² by the divisor.
x²(x − 1) = x³ − x²
Subtract.
(x³ + 2x² − 4x + 1) − (x³ − x²)
3x² − 4x + 1
Divide the leading terms.
3x² ÷ x = 3x
Multiply 3x by the divisor.
3x(x − 1) = 3x² − 3x
Subtract.
(3x² − 4x + 1) − (3x² − 3x)
−x + 1
Divide the leading terms.
−x ÷ x = −1
Multiply −1 by the divisor.
−1(x − 1) = −x + 1
Subtract.
(−x + 1) − (−x + 1)
0
Quotient: x² + 3x − 1
Remainder: 0
Answer: x² + 3x − 1
Divide x⁴ − 6x³ + 9x² + 4x − 12 by x − 2.
Divide the leading terms.
x⁴ ÷ x = x³
Multiply x³ by the divisor.
x³(x − 2) = x⁴ − 2x³
Subtract.
(x⁴ − 6x³ + 9x² + 4x − 12) − (x⁴ − 2x³)
−4x³ + 9x² + 4x − 12
Divide the leading terms.
−4x³ ÷ x = −4x²
Multiply −4x² by the divisor.
−4x²(x − 2) = −4x³ + 8x²
Subtract.
(−4x³ + 9x² + 4x − 12) − (−4x³ + 8x²)
x² + 4x − 12
Divide the leading terms.
x² ÷ x = x
Multiply x by the divisor.
x(x − 2) = x² − 2x
Subtract.
(x² + 4x − 12) − (x² − 2x)
6x − 12
Divide the leading terms.
6x ÷ x = 6
Multiply 6 by the divisor.
6(x − 2) = 6x − 12
Subtract.
(6x − 12) − (6x − 12)
0
Quotient: x³ − 4x² + x + 6
Remainder: 0
Answer: x³ − 4x² + x + 6
Divide x³ − 2x² − 5x + 6 by x − 3.
Divide the leading terms.
x³ ÷ x = x²
Multiply x² by the divisor.
x²(x − 3) = x³ − 3x²
Subtract.
(x³ − 2x² − 5x + 6) − (x³ − 3x²)
x² − 5x + 6
Divide the leading terms.
x² ÷ x = x
Multiply x by the divisor.
x(x − 3) = x² − 3x
Subtract.
(x² − 5x + 6) − (x² − 3x)
−2x + 6
Divide the leading terms.
−2x ÷ x = −2
Multiply −2 by the divisor.
−2(x − 3) = −2x + 6
Subtract.
(−2x + 6) − (−2x + 6)
0
Quotient: x² + x − 2
Remainder: 0
Answer: x² + x − 2
Divide x⁴ − x³ − 8x² + 4x + 16 by x + 2.
Divide the leading terms.
x⁴ ÷ x = x³
Multiply x³ by the divisor.
x³(x + 2) = x⁴ + 2x³
Subtract.
(x⁴ − x³ − 8x² + 4x + 16) − (x⁴ + 2x³)
−3x³ − 8x² + 4x + 16
Divide the leading terms.
−3x³ ÷ x = −3x²
Multiply −3x² by the divisor.
−3x²(x + 2) = −3x³ − 6x²
Subtract.
(−3x³ − 8x² + 4x + 16) − (−3x³ − 6x²)
−2x² + 4x + 16
Divide the leading terms.
−2x² ÷ x = −2x
Multiply −2x by the divisor.
−2x(x + 2) = −2x² − 4x
Subtract.
(−2x² + 4x + 16) − (−2x² − 4x)
8x + 16
Divide the leading terms.
8x ÷ x = 8
Multiply 8 by the divisor.
8(x + 2) = 8x + 16
Subtract.
(8x + 16) − (8x + 16)
0
Quotient: x³ − 3x² − 2x + 8
Remainder: 0
Answer: x³ − 3x² − 2x + 8
Divide x³ + 2x² − 5x − 6 by x + 1.
Divide the leading terms.
x³ ÷ x = x²
Multiply x² by the divisor.
x²(x + 1) = x³ + x²
Subtract.
(x³ + 2x² − 5x − 6) − (x³ + x²)
x² − 5x − 6
Divide the leading terms.
x² ÷ x = x
Multiply x by the divisor.
x(x + 1) = x² + x
Subtract.
(x² − 5x − 6) − (x² + x)
−6x − 6
Divide the leading terms.
−6x ÷ x = −6
Multiply −6 by the divisor.
−6(x + 1) = −6x − 6
Subtract.
(−6x − 6) − (−6x − 6)
0
Quotient: x² + x − 6
Remainder: 0
Answer: x² + x − 6
Divide x⁴ − 3x³ − 3x² + 8x − 3 by x − 1.
Divide the leading terms.
x⁴ ÷ x = x³
Multiply x³ by the divisor.
x³(x − 1) = x⁴ − x³
Subtract.
(x⁴ − 3x³ − 3x² + 8x − 3) − (x⁴ − x³)
−2x³ − 3x² + 8x − 3
Divide the leading terms.
−2x³ ÷ x = −2x²
Multiply −2x² by the divisor.
−2x²(x − 1) = −2x³ + 2x²
Subtract.
(−2x³ − 3x² + 8x − 3) − (−2x³ + 2x²)
−5x² + 8x − 3
Divide the leading terms.
−5x² ÷ x = −5x
Multiply −5x by the divisor.
−5x(x − 1) = −5x² + 5x
Subtract.
(−5x² + 8x − 3) − (−5x² + 5x)
3x − 3
Divide the leading terms.
3x ÷ x = 3
Multiply 3 by the divisor.
3(x − 1) = 3x − 3
Subtract.
(3x − 3) − (3x − 3)
0
Quotient: x³ − 2x² − 5x + 3
Remainder: 0
Answer: x³ − 2x² − 5x + 3
There is a remainder. Write any missing power as 0.
Divide x³ − 2x² + x + 1 by x − 2.
Divide the leading terms.
x³ ÷ x = x²
Multiply x² by the divisor.
x²(x − 2) = x³ − 2x²
Subtract.
(x³ − 2x² + x + 1) − (x³ − 2x²)
x + 1
Divide the leading terms.
x ÷ x = 1
Multiply 1 by the divisor.
1(x − 2) = x − 2
Subtract.
(x + 1) − (x − 2)
3
Quotient: x² + 1
Remainder: 3
Answer: x² + 1 + 3x − 2
Divide x³ + 3x − 1 by x + 1.
Write each missing power as 0 times that power before you subtract.
Divide the leading terms.
x³ ÷ x = x²
Multiply x² by the divisor.
x²(x + 1) = x³ + x²
Subtract.
(x³ + 3x − 1) − (x³ + x²)
−x² + 3x − 1
Divide the leading terms.
−x² ÷ x = −x
Multiply −x by the divisor.
−x(x + 1) = −x² − x
Subtract.
(−x² + 3x − 1) − (−x² − x)
4x − 1
Divide the leading terms.
4x ÷ x = 4
Multiply 4 by the divisor.
4(x + 1) = 4x + 4
Subtract.
(4x − 1) − (4x + 4)
−5
Quotient: x² − x + 4
Remainder: −5
Answer: x² − x + 4 + (−5)x + 1
Divide x⁴ − x³ − 6x² − 2x + 10 by x − 3.
Divide the leading terms.
x⁴ ÷ x = x³
Multiply x³ by the divisor.
x³(x − 3) = x⁴ − 3x³
Subtract.
(x⁴ − x³ − 6x² − 2x + 10) − (x⁴ − 3x³)
2x³ − 6x² − 2x + 10
Divide the leading terms.
2x³ ÷ x = 2x²
Multiply 2x² by the divisor.
2x²(x − 3) = 2x³ − 6x²
Subtract.
(2x³ − 6x² − 2x + 10) − (2x³ − 6x²)
−2x + 10
Divide the leading terms.
−2x ÷ x = −2
Multiply −2 by the divisor.
−2(x − 3) = −2x + 6
Subtract.
(−2x + 10) − (−2x + 6)
4
Quotient: x³ + 2x² − 2
Remainder: 4
Answer: x³ + 2x² − 2 + 4x − 3
Divide x⁴ + 2x³ + 3x² + 7x + 1 by x + 2.
Divide the leading terms.
x⁴ ÷ x = x³
Multiply x³ by the divisor.
x³(x + 2) = x⁴ + 2x³
Subtract.
(x⁴ + 2x³ + 3x² + 7x + 1) − (x⁴ + 2x³)
3x² + 7x + 1
Divide the leading terms.
3x² ÷ x = 3x
Multiply 3x by the divisor.
3x(x + 2) = 3x² + 6x
Subtract.
(3x² + 7x + 1) − (3x² + 6x)
x + 1
Divide the leading terms.
x ÷ x = 1
Multiply 1 by the divisor.
1(x + 2) = x + 2
Subtract.
(x + 1) − (x + 2)
−1
Quotient: x³ + 3x + 1
Remainder: −1
Answer: x³ + 3x + 1 + (−1)x + 2
Divide x³ − 4x² + 6x − 22 by x − 4.
Divide the leading terms.
x³ ÷ x = x²
Multiply x² by the divisor.
x²(x − 4) = x³ − 4x²
Subtract.
(x³ − 4x² + 6x − 22) − (x³ − 4x²)
6x − 22
Divide the leading terms.
6x ÷ x = 6
Multiply 6 by the divisor.
6(x − 4) = 6x − 24
Subtract.
(6x − 22) − (6x − 24)
2
Quotient: x² + 6
Remainder: 2
Answer: x² + 6 + 2x − 4
Divide 2x⁴ + 2x³ + x² − 2x + 2 by x + 1.
Divide the leading terms.
2x⁴ ÷ x = 2x³
Multiply 2x³ by the divisor.
2x³(x + 1) = 2x⁴ + 2x³
Subtract.
(2x⁴ + 2x³ + x² − 2x + 2) − (2x⁴ + 2x³)
x² − 2x + 2
Divide the leading terms.
x² ÷ x = x
Multiply x by the divisor.
x(x + 1) = x² + x
Subtract.
(x² − 2x + 2) − (x² + x)
−3x + 2
Divide the leading terms.
−3x ÷ x = −3
Multiply −3 by the divisor.
−3(x + 1) = −3x − 3
Subtract.
(−3x + 2) − (−3x − 3)
5
Quotient: 2x³ + x − 3
Remainder: 5
Answer: 2x³ + x − 3 + 5x + 1
The divisor is a quadratic.
Divide 2x³ + x² + 5x + 5 by x² − x + 2.
Divide the leading terms.
2x³ ÷ x² = 2x
Multiply 2x by the divisor.
2x(x² − x + 2) = 2x³ − 2x² + 4x
Subtract.
(2x³ + x² + 5x + 5) − (2x³ − 2x² + 4x)
3x² + x + 5
Divide the leading terms.
3x² ÷ x² = 3
Multiply 3 by the divisor.
3(x² − x + 2) = 3x² − 3x + 6
Subtract.
(3x² + x + 5) − (3x² − 3x + 6)
4x − 1
Quotient: 2x + 3
Remainder: 4x − 1
Answer: 2x + 3 + (4x − 1)x² − x + 2
Divide x³ + 2x² − 3x + 1 by x² + x − 2.
Divide the leading terms.
x³ ÷ x² = x
Multiply x by the divisor.
x(x² + x − 2) = x³ + x² − 2x
Subtract.
(x³ + 2x² − 3x + 1) − (x³ + x² − 2x)
x² − x + 1
Divide the leading terms.
x² ÷ x² = 1
Multiply 1 by the divisor.
1(x² + x − 2) = x² + x − 2
Subtract.
(x² − x + 1) − (x² + x − 2)
−2x + 3
Quotient: x + 1
Remainder: −2x + 3
Answer: x + 1 + (−2x + 3)x² + x − 2
Divide 2x³ − 4x² + 2x + 1 by x² + 1.
Divide the leading terms.
2x³ ÷ x² = 2x
Multiply 2x by the divisor.
2x(x² + 1) = 2x³ + 2x
Subtract.
(2x³ − 4x² + 2x + 1) − (2x³ + 2x)
−4x² + 1
Divide the leading terms.
−4x² ÷ x² = −4
Multiply −4 by the divisor.
−4(x² + 1) = −4x² − 4
Subtract.
(−4x² + 1) − (−4x² − 4)
5
Quotient: 2x − 4
Remainder: 5
Answer: 2x − 4 + 5x² + 1
Divide 3x³ + 8x² − 2x − 1 by x² + 2x − 1.
Divide the leading terms.
3x³ ÷ x² = 3x
Multiply 3x by the divisor.
3x(x² + 2x − 1) = 3x³ + 6x² − 3x
Subtract.
(3x³ + 8x² − 2x − 1) − (3x³ + 6x² − 3x)
2x² + x − 1
Divide the leading terms.
2x² ÷ x² = 2
Multiply 2 by the divisor.
2(x² + 2x − 1) = 2x² + 4x − 2
Subtract.
(2x² + x − 1) − (2x² + 4x − 2)
−3x + 1
Quotient: 3x + 2
Remainder: −3x + 1
Answer: 3x + 2 + (−3x + 1)x² + 2x − 1
Divide 4x³ + 3x² + 17x − 5 by x² + x + 3.
Divide the leading terms.
4x³ ÷ x² = 4x
Multiply 4x by the divisor.
4x(x² + x + 3) = 4x³ + 4x² + 12x
Subtract.
(4x³ + 3x² + 17x − 5) − (4x³ + 4x² + 12x)
−x² + 5x − 5
Divide the leading terms.
−x² ÷ x² = −1
Multiply −1 by the divisor.
−1(x² + x + 3) = −x² − x − 3
Subtract.
(−x² + 5x − 5) − (−x² − x − 3)
6x − 2
Quotient: 4x − 1
Remainder: 6x − 2
Answer: 4x − 1 + (6x − 2)x² + x + 3
Divide 2x³ + 9x² − 7x − 11 by 2x² − x − 3.
Divide the leading terms.
2x³ ÷ 2x² = x
Multiply x by the divisor.
x(2x² − x − 3) = 2x³ − x² − 3x
Subtract.
(2x³ + 9x² − 7x − 11) − (2x³ − x² − 3x)
10x² − 4x − 11
Divide the leading terms.
10x² ÷ 2x² = 5
Multiply 5 by the divisor.
5(2x² − x − 3) = 10x² − 5x − 15
Subtract.
(10x² − 4x − 11) − (10x² − 5x − 15)
x + 4
Quotient: x + 5
Remainder: x + 4
Answer: x + 5 + (x + 4)2x² − x − 3
Need help with polynomial long division?
Book a Session with Amy