M
Math Coach Amy
Amy Ferguson Moncure

H Algebra 2/Trig · Unit 3

Polynomial Area

Divide the area polynomial by the length to find the other side.

Jump to section:

Key Ideas Easy Medium Hard

Area = length × width

Width = area ÷ length

What to Remember

Area = length × width

Divide the area by the length. The remainder is 0 when the length matches the area.

Level 1

Easy

The width is linear.

Example
A rectangle has area x² + 8x + 15 and length x + 5. Find the width.

The width is the area divided by the length.

Divide x² + 8x + 15 by x + 5.

Divide the leading terms.

x² ÷ x = x

Multiply x by the divisor.

x(x + 5) = x² + 5x

Subtract.

(x² + 8x + 15) − (x² + 5x)

3x + 15

Divide the leading terms.

3x ÷ x = 3

Multiply 3 by the divisor.

3(x + 5) = 3x + 15

Subtract.

(3x + 15) − (3x + 15)

0

Quotient: x + 3

Remainder: 0

Answer: x + 3

1
A rectangle has area x² + 7x + 12 and length x + 3. Find the width.
Show Answer

The width is the area divided by the length.

Divide x² + 7x + 12 by x + 3.

Divide the leading terms.

x² ÷ x = x

Multiply x by the divisor.

x(x + 3) = x² + 3x

Subtract.

(x² + 7x + 12) − (x² + 3x)

4x + 12

Divide the leading terms.

4x ÷ x = 4

Multiply 4 by the divisor.

4(x + 3) = 4x + 12

Subtract.

(4x + 12) − (4x + 12)

0

Quotient: x + 4

Remainder: 0

Answer: x + 4

2
A rectangle has area x² − x − 12 and length x − 4. Find the width.
Show Answer

The width is the area divided by the length.

Divide x² − x − 12 by x − 4.

Divide the leading terms.

x² ÷ x = x

Multiply x by the divisor.

x(x − 4) = x² − 4x

Subtract.

(x² − x − 12) − (x² − 4x)

3x − 12

Divide the leading terms.

3x ÷ x = 3

Multiply 3 by the divisor.

3(x − 4) = 3x − 12

Subtract.

(3x − 12) − (3x − 12)

0

Quotient: x + 3

Remainder: 0

Answer: x + 3

3
A rectangle has area x² + 2x − 15 and length x + 5. Find the width.
Show Answer

The width is the area divided by the length.

Divide x² + 2x − 15 by x + 5.

Divide the leading terms.

x² ÷ x = x

Multiply x by the divisor.

x(x + 5) = x² + 5x

Subtract.

(x² + 2x − 15) − (x² + 5x)

−3x − 15

Divide the leading terms.

−3x ÷ x = −3

Multiply −3 by the divisor.

−3(x + 5) = −3x − 15

Subtract.

(−3x − 15) − (−3x − 15)

0

Quotient: x − 3

Remainder: 0

Answer: x − 3

4
A rectangle has area x² − 9x + 20 and length x − 4. Find the width.
Show Answer

The width is the area divided by the length.

Divide x² − 9x + 20 by x − 4.

Divide the leading terms.

x² ÷ x = x

Multiply x by the divisor.

x(x − 4) = x² − 4x

Subtract.

(x² − 9x + 20) − (x² − 4x)

−5x + 20

Divide the leading terms.

−5x ÷ x = −5

Multiply −5 by the divisor.

−5(x − 4) = −5x + 20

Subtract.

(−5x + 20) − (−5x + 20)

0

Quotient: x − 5

Remainder: 0

Answer: x − 5

5
A rectangle has area 2x² + 7x + 3 and length x + 3. Find the width.
Show Answer

The width is the area divided by the length.

Divide 2x² + 7x + 3 by x + 3.

Divide the leading terms.

2x² ÷ x = 2x

Multiply 2x by the divisor.

2x(x + 3) = 2x² + 6x

Subtract.

(2x² + 7x + 3) − (2x² + 6x)

x + 3

Divide the leading terms.

x ÷ x = 1

Multiply 1 by the divisor.

1(x + 3) = x + 3

Subtract.

(x + 3) − (x + 3)

0

Quotient: 2x + 1

Remainder: 0

Answer: 2x + 1

Level 2

Medium

The length has a coefficient greater than 1.

Example
A rectangle has area 3x² + 11x − 4 and length 3x − 1. Find the width.

The width is the area divided by the length.

Divide 3x² + 11x − 4 by 3x − 1.

Divide the leading terms.

3x² ÷ 3x = x

Multiply x by the divisor.

x(3x − 1) = 3x² − x

Subtract.

(3x² + 11x − 4) − (3x² − x)

12x − 4

Divide the leading terms.

12x ÷ 3x = 4

Multiply 4 by the divisor.

4(3x − 1) = 12x − 4

Subtract.

(12x − 4) − (12x − 4)

0

Quotient: x + 4

Remainder: 0

Answer: x + 4

1
A rectangle has area 2x² + 5x − 3 and length 2x − 1. Find the width.
Show Answer

The width is the area divided by the length.

Divide 2x² + 5x − 3 by 2x − 1.

Divide the leading terms.

2x² ÷ 2x = x

Multiply x by the divisor.

x(2x − 1) = 2x² − x

Subtract.

(2x² + 5x − 3) − (2x² − x)

6x − 3

Divide the leading terms.

6x ÷ 2x = 3

Multiply 3 by the divisor.

3(2x − 1) = 6x − 3

Subtract.

(6x − 3) − (6x − 3)

0

Quotient: x + 3

Remainder: 0

Answer: x + 3

2
A rectangle has area 3x² − 10x − 8 and length 3x + 2. Find the width.
Show Answer

The width is the area divided by the length.

Divide 3x² − 10x − 8 by 3x + 2.

Divide the leading terms.

3x² ÷ 3x = x

Multiply x by the divisor.

x(3x + 2) = 3x² + 2x

Subtract.

(3x² − 10x − 8) − (3x² + 2x)

−12x − 8

Divide the leading terms.

−12x ÷ 3x = −4

Multiply −4 by the divisor.

−4(3x + 2) = −12x − 8

Subtract.

(−12x − 8) − (−12x − 8)

0

Quotient: x − 4

Remainder: 0

Answer: x − 4

3
A rectangle has area 4x² + 4x − 15 and length 2x − 3. Find the width.
Show Answer

The width is the area divided by the length.

Divide 4x² + 4x − 15 by 2x − 3.

Divide the leading terms.

4x² ÷ 2x = 2x

Multiply 2x by the divisor.

2x(2x − 3) = 4x² − 6x

Subtract.

(4x² + 4x − 15) − (4x² − 6x)

10x − 15

Divide the leading terms.

10x ÷ 2x = 5

Multiply 5 by the divisor.

5(2x − 3) = 10x − 15

Subtract.

(10x − 15) − (10x − 15)

0

Quotient: 2x + 5

Remainder: 0

Answer: 2x + 5

4
A rectangle has area 6x² − 7x − 3 and length 2x − 3. Find the width.
Show Answer

The width is the area divided by the length.

Divide 6x² − 7x − 3 by 2x − 3.

Divide the leading terms.

6x² ÷ 2x = 3x

Multiply 3x by the divisor.

3x(2x − 3) = 6x² − 9x

Subtract.

(6x² − 7x − 3) − (6x² − 9x)

2x − 3

Divide the leading terms.

2x ÷ 2x = 1

Multiply 1 by the divisor.

1(2x − 3) = 2x − 3

Subtract.

(2x − 3) − (2x − 3)

0

Quotient: 3x + 1

Remainder: 0

Answer: 3x + 1

5
A rectangle has area 6x² + x − 2 and length 3x + 2. Find the width.
Show Answer

The width is the area divided by the length.

Divide 6x² + x − 2 by 3x + 2.

Divide the leading terms.

6x² ÷ 3x = 2x

Multiply 2x by the divisor.

2x(3x + 2) = 6x² + 4x

Subtract.

(6x² + x − 2) − (6x² + 4x)

−3x − 2

Divide the leading terms.

−3x ÷ 3x = −1

Multiply −1 by the divisor.

−1(3x + 2) = −3x − 2

Subtract.

(−3x − 2) − (−3x − 2)

0

Quotient: 2x − 1

Remainder: 0

Answer: 2x − 1

Level 3

Hard

The area is a cubic, so the width is a quadratic.

Example
A rectangle has area x³ + x² − 5x − 2 and length x − 2. Find the width.

The width is the area divided by the length.

Divide x³ + x² − 5x − 2 by x − 2.

Divide the leading terms.

x³ ÷ x = x²

Multiply x² by the divisor.

x²(x − 2) = x³ − 2x²

Subtract.

(x³ + x² − 5x − 2) − (x³ − 2x²)

3x² − 5x − 2

Divide the leading terms.

3x² ÷ x = 3x

Multiply 3x by the divisor.

3x(x − 2) = 3x² − 6x

Subtract.

(3x² − 5x − 2) − (3x² − 6x)

x − 2

Divide the leading terms.

x ÷ x = 1

Multiply 1 by the divisor.

1(x − 2) = x − 2

Subtract.

(x − 2) − (x − 2)

0

Quotient: x² + 3x + 1

Remainder: 0

Answer: x² + 3x + 1

1
A rectangle has area x³ − x² + 2x + 4 and length x + 1. Find the width.
Show Answer

The width is the area divided by the length.

Divide x³ − x² + 2x + 4 by x + 1.

Divide the leading terms.

x³ ÷ x = x²

Multiply x² by the divisor.

x²(x + 1) = x³ + x²

Subtract.

(x³ − x² + 2x + 4) − (x³ + x²)

−2x² + 2x + 4

Divide the leading terms.

−2x² ÷ x = −2x

Multiply −2x by the divisor.

−2x(x + 1) = −2x² − 2x

Subtract.

(−2x² + 2x + 4) − (−2x² − 2x)

4x + 4

Divide the leading terms.

4x ÷ x = 4

Multiply 4 by the divisor.

4(x + 1) = 4x + 4

Subtract.

(4x + 4) − (4x + 4)

0

Quotient: x² − 2x + 4

Remainder: 0

Answer: x² − 2x + 4

2
A rectangle has area 2x³ + x² + 5x − 3 and length 2x − 1. Find the width.
Show Answer

The width is the area divided by the length.

Divide 2x³ + x² + 5x − 3 by 2x − 1.

Divide the leading terms.

2x³ ÷ 2x = x²

Multiply x² by the divisor.

x²(2x − 1) = 2x³ − x²

Subtract.

(2x³ + x² + 5x − 3) − (2x³ − x²)

2x² + 5x − 3

Divide the leading terms.

2x² ÷ 2x = x

Multiply x by the divisor.

x(2x − 1) = 2x² − x

Subtract.

(2x² + 5x − 3) − (2x² − x)

6x − 3

Divide the leading terms.

6x ÷ 2x = 3

Multiply 3 by the divisor.

3(2x − 1) = 6x − 3

Subtract.

(6x − 3) − (6x − 3)

0

Quotient: x² + x + 3

Remainder: 0

Answer: x² + x + 3

3
A rectangle has area 2x³ − 5x² + x − 12 and length x − 3. Find the width.
Show Answer

The width is the area divided by the length.

Divide 2x³ − 5x² + x − 12 by x − 3.

Divide the leading terms.

2x³ ÷ x = 2x²

Multiply 2x² by the divisor.

2x²(x − 3) = 2x³ − 6x²

Subtract.

(2x³ − 5x² + x − 12) − (2x³ − 6x²)

x² + x − 12

Divide the leading terms.

x² ÷ x = x

Multiply x by the divisor.

x(x − 3) = x² − 3x

Subtract.

(x² + x − 12) − (x² − 3x)

4x − 12

Divide the leading terms.

4x ÷ x = 4

Multiply 4 by the divisor.

4(x − 3) = 4x − 12

Subtract.

(4x − 12) − (4x − 12)

0

Quotient: 2x² + x + 4

Remainder: 0

Answer: 2x² + x + 4

4
A rectangle has area x³ + 3x² − 2x + 8 and length x + 4. Find the width.
Show Answer

The width is the area divided by the length.

Divide x³ + 3x² − 2x + 8 by x + 4.

Divide the leading terms.

x³ ÷ x = x²

Multiply x² by the divisor.

x²(x + 4) = x³ + 4x²

Subtract.

(x³ + 3x² − 2x + 8) − (x³ + 4x²)

−x² − 2x + 8

Divide the leading terms.

−x² ÷ x = −x

Multiply −x by the divisor.

−x(x + 4) = −x² − 4x

Subtract.

(−x² − 2x + 8) − (−x² − 4x)

2x + 8

Divide the leading terms.

2x ÷ x = 2

Multiply 2 by the divisor.

2(x + 4) = 2x + 8

Subtract.

(2x + 8) − (2x + 8)

0

Quotient: x² − x + 2

Remainder: 0

Answer: x² − x + 2

5
A rectangle has area 3x³ − 4x² − x + 2 and length 3x + 2. Find the width.
Show Answer

The width is the area divided by the length.

Divide 3x³ − 4x² − x + 2 by 3x + 2.

Divide the leading terms.

3x³ ÷ 3x = x²

Multiply x² by the divisor.

x²(3x + 2) = 3x³ + 2x²

Subtract.

(3x³ − 4x² − x + 2) − (3x³ + 2x²)

−6x² − x + 2

Divide the leading terms.

−6x² ÷ 3x = −2x

Multiply −2x by the divisor.

−2x(3x + 2) = −6x² − 4x

Subtract.

(−6x² − x + 2) − (−6x² − 4x)

3x + 2

Divide the leading terms.

3x ÷ 3x = 1

Multiply 1 by the divisor.

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

Subtract.

(3x + 2) − (3x + 2)

0

Quotient: x² − 2x + 1

Remainder: 0

Answer: x² − 2x + 1

Need help with a polynomial area?

Book a Session with Amy