Solve Polynomial Equations

Jump to section:

Key Ideas Easy Medium Hard

Even powers: let u = x².

Four terms: group, then factor.

What to Remember

A fourth power with only even powers

Let u = x². Factor the quadratic in u, then take square roots.

Four terms

Group two and two, factor each group, then set each factor equal to 0.

Level 1

Easy

Every zero is a whole number.

Example

Solve. x⁴ − 17x² + 16 = 0

x⁴ − 17x² + 16 = 0

Let u = x².

u² − 17u + 16 = 0

(u − 16)(u − 1) = 0

u = 16 or u = 1

Put x² back in for u.

x² = 16

x = 4 or x = −4

x² = 1

x = 1 or x = −1

Answer: x = 4 or x = −4
x = 1 or x = −1

Solve. x⁴ − 10x² + 9 = 0

Show Answer

x⁴ − 10x² + 9 = 0

Let u = x².

u² − 10u + 9 = 0

(u − 9)(u − 1) = 0

u = 9 or u = 1

Put x² back in for u.

x² = 9

x = 3 or x = −3

x² = 1

x = 1 or x = −1

Answer: x = 3 or x = −3
x = 1 or x = −1

Solve. x⁴ − 13x² + 36 = 0

Show Answer

x⁴ − 13x² + 36 = 0

Let u = x².

u² − 13u + 36 = 0

(u − 9)(u − 4) = 0

u = 9 or u = 4

Put x² back in for u.

x² = 9

x = 3 or x = −3

x² = 4

x = 2 or x = −2

Answer: x = 3 or x = −3
x = 2 or x = −2

Solve. x⁴ − 20x² + 64 = 0

Show Answer

x⁴ − 20x² + 64 = 0

Let u = x².

u² − 20u + 64 = 0

(u − 16)(u − 4) = 0

u = 16 or u = 4

Put x² back in for u.

x² = 16

x = 4 or x = −4

x² = 4

x = 2 or x = −2

Answer: x = 4 or x = −4
x = 2 or x = −2

Solve. x⁴ − 5x² + 4 = 0

Show Answer

x⁴ − 5x² + 4 = 0

Let u = x².

u² − 5u + 4 = 0

(u − 4)(u − 1) = 0

u = 4 or u = 1

Put x² back in for u.

x² = 4

x = 2 or x = −2

x² = 1

x = 1 or x = −1

Answer: x = 2 or x = −2
x = 1 or x = −1

Solve. x⁴ − 25x² + 144 = 0

Show Answer

x⁴ − 25x² + 144 = 0

Let u = x².

u² − 25u + 144 = 0

(u − 16)(u − 9) = 0

u = 16 or u = 9

Put x² back in for u.

x² = 16

x = 4 or x = −4

x² = 9

x = 3 or x = −3

Answer: x = 4 or x = −4
x = 3 or x = −3

Level 2

Medium

Group, then use a square root.

Example

Solve by grouping. x³ + 4x² − 3x − 12 = 0

x³ + 4x² − 3x − 12 = 0

Group the first two terms and the last two terms.

x²(x + 4) − 3(x + 4) = 0

(x² − 3)(x + 4) = 0

x + 4 = 0

x = −4

x² − 3 = 0

x² = 3

x = square root of 3 or x = −square root of 3

Answer: x = −4
x = square root of 3 or x = −square root of 3

Solve by grouping. x³ − 6x² − 5x + 30 = 0

Show Answer

x³ − 6x² − 5x + 30 = 0

Group the first two terms and the last two terms.

x²(x − 6) − 5(x − 6) = 0

(x² − 5)(x − 6) = 0

x − 6 = 0

x = 6

x² − 5 = 0

x² = 5

x = square root of 5 or x = −square root of 5

Answer: x = 6
x = square root of 5 or x = −square root of 5

Solve by grouping. x³ + 2x² − 7x − 14 = 0

Show Answer

x³ + 2x² − 7x − 14 = 0

Group the first two terms and the last two terms.

x²(x + 2) − 7(x + 2) = 0

(x² − 7)(x + 2) = 0

x + 2 = 0

x = −2

x² − 7 = 0

x² = 7

x = square root of 7 or x = −square root of 7

Answer: x = −2
x = square root of 7 or x = −square root of 7

Solve by grouping. x³ − 5x² − 2x + 10 = 0

Show Answer

x³ − 5x² − 2x + 10 = 0

Group the first two terms and the last two terms.

x²(x − 5) − 2(x − 5) = 0

(x² − 2)(x − 5) = 0

x − 5 = 0

x = 5

x² − 2 = 0

x² = 2

x = square root of 2 or x = −square root of 2

Answer: x = 5
x = square root of 2 or x = −square root of 2

Solve by grouping. 3x³ − 3x² − 6x + 6 = 0

Show Answer

3x³ − 3x² − 6x + 6 = 0

Group the first two terms and the last two terms.

3x²(x − 1) − 6(x − 1) = 0

(3x² − 6)(x − 1) = 0

Factor 3 from the quadratic.

3(x² − 2)(x − 1) = 0

x − 1 = 0

x = 1

x² − 2 = 0

x² = 2

x = square root of 2 or x = −square root of 2

Answer: x = 1
x = square root of 2 or x = −square root of 2

Solve by grouping. 2x³ + 8x² − 6x − 24 = 0

Show Answer

2x³ + 8x² − 6x − 24 = 0

Group the first two terms and the last two terms.

2x²(x + 4) − 6(x + 4) = 0

(2x² − 6)(x + 4) = 0

Factor 2 from the quadratic.

2(x² − 3)(x + 4) = 0

x + 4 = 0

x = −4

x² − 3 = 0

x² = 3

x = square root of 3 or x = −square root of 3

Answer: x = −4
x = square root of 3 or x = −square root of 3

Level 3

Hard

One factor is imaginary, or a square root needs rationalizing.

Example

Solve. x⁴ − 8x² − 9 = 0

x⁴ − 8x² − 9 = 0

Let u = x².

u² − 8u − 9 = 0

(u − 9)(u + 1) = 0

u = 9 or u = −1

Put x² back in for u.

x² = 9

x = 3 or x = −3

x² = −1

A negative under the square root gives an imaginary number.

x = i or x = −i

Answer: x = 3 or x = −3
x = i or x = −i

Solve. x⁴ − 2x² − 8 = 0

Show Answer

x⁴ − 2x² − 8 = 0

Let u = x².

u² − 2u − 8 = 0

(u − 4)(u + 2) = 0

u = 4 or u = −2

Put x² back in for u.

x² = 4

x = 2 or x = −2

x² = −2

A negative under the square root gives an imaginary number.

x = isquare root of 2 or x = −isquare root of 2

Answer: x = 2 or x = −2
x = isquare root of 2 or x = −isquare root of 2

Solve. x⁴ + 3x² − 4 = 0

Show Answer

x⁴ + 3x² − 4 = 0

Let u = x².

u² + 3u − 4 = 0

(u + 4)(u − 1) = 0

u = −4 or u = 1

Put x² back in for u.

x² = −4

A negative under the square root gives an imaginary number.

x = 2i or x = −2i

x² = 1

x = 1 or x = −1

Answer: x = 2i or x = −2i
x = 1 or x = −1

Solve. x⁴ − 7x² − 18 = 0

Show Answer

x⁴ − 7x² − 18 = 0

Let u = x².

u² − 7u − 18 = 0

(u − 9)(u + 2) = 0

u = 9 or u = −2

Put x² back in for u.

x² = 9

x = 3 or x = −3

x² = −2

A negative under the square root gives an imaginary number.

x = isquare root of 2 or x = −isquare root of 2

Answer: x = 3 or x = −3
x = isquare root of 2 or x = −isquare root of 2

Solve. x⁴ + 5x² − 36 = 0

Show Answer

x⁴ + 5x² − 36 = 0

Let u = x².

u² + 5u − 36 = 0

(u + 9)(u − 4) = 0

u = −9 or u = 4

Put x² back in for u.

x² = −9

A negative under the square root gives an imaginary number.

x = 3i or x = −3i

x² = 4

x = 2 or x = −2

Answer: x = 3i or x = −3i
x = 2 or x = −2

Solve by grouping. 3x³ + 6x² − 5x − 10 = 0

Show Answer

3x³ + 6x² − 5x − 10 = 0

Group the first two terms and the last two terms.

3x²(x + 2) − 5(x + 2) = 0

(3x² − 5)(x + 2) = 0

x + 2 = 0

x = −2

3x² − 5 = 0

3x² = 5

Divide both sides by 3.

x² = 53

Rationalize the denominator.

x = square root of 153 or x = −square root of 153

Answer: x = −2
x = square root of 153 or x = −square root of 153

Need help solving a polynomial equation?

Book a Session with Amy