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Amy Ferguson Moncure

H Algebra 2/Trig · Unit 3

Polynomial from Zeros

Include each missing conjugate, then write the least-degree polynomial in factored form.

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Key Ideas Easy Medium Hard

Square-root zeros come in ± pairs.

Imaginary zeros come in conjugate pairs.

What to Remember

Conjugate pairs

Real coefficients: a zero of a + bi brings a − bi, and a zero with a square root brings the same number with the opposite square root.

Factored form

An integer zero c gives (x − c). A fraction zero uses a factor with integer coefficients, such as x = −3/2 giving (2x + 3).

Each conjugate pair becomes a quadratic factor.

Level 1

Easy

One integer, one square-root pair, and one imaginary pair.

Example
Write a polynomial of least degree with real coefficients that has these zeros. Leave it in factored form. x = 2 x = square root of 5 x = 1 − i

A polynomial with real coefficients includes the conjugate of each square-root zero and each imaginary zero.

x = 2 gives the factor (x − 2).

square root of 5 and −square root of 5 are both zeros.

(x − square root of 5)(x + square root of 5) = x² − 5

1 − i and 1 + i are both zeros.

(x − (1 − i))(x − (1 + i))

= (x − 1)² − (i)²

i² = −1, so (i)² = −1.

= (x − 1)² + 1

= x² − 2x + 2

y = (x − 2)(x² − 5)(x² − 2x + 2)

Answer: y = (x − 2)(x² − 5)(x² − 2x + 2)

1
Write a polynomial of least degree with real coefficients that has these zeros. Leave it in factored form. x = 3 x = square root of 2 x = 2 − i
Show Answer

A polynomial with real coefficients includes the conjugate of each square-root zero and each imaginary zero.

x = 3 gives the factor (x − 3).

square root of 2 and −square root of 2 are both zeros.

(x − square root of 2)(x + square root of 2) = x² − 2

2 − i and 2 + i are both zeros.

(x − (2 − i))(x − (2 + i))

= (x − 2)² − (i)²

i² = −1, so (i)² = −1.

= (x − 2)² + 1

= x² − 4x + 5

y = (x − 3)(x² − 2)(x² − 4x + 5)

Answer: y = (x − 3)(x² − 2)(x² − 4x + 5)

2
Write a polynomial of least degree with real coefficients that has these zeros. Leave it in factored form. x = −1 x = square root of 7 x = 4 + i
Show Answer

A polynomial with real coefficients includes the conjugate of each square-root zero and each imaginary zero.

x = −1 gives the factor (x + 1).

square root of 7 and −square root of 7 are both zeros.

(x − square root of 7)(x + square root of 7) = x² − 7

4 + i and 4 − i are both zeros.

(x − (4 + i))(x − (4 − i))

= (x − 4)² − (i)²

i² = −1, so (i)² = −1.

= (x − 4)² + 1

= x² − 8x + 17

y = (x + 1)(x² − 7)(x² − 8x + 17)

Answer: y = (x + 1)(x² − 7)(x² − 8x + 17)

3
Write a polynomial of least degree with real coefficients that has these zeros. Leave it in factored form. x = 6 x = square root of 6 x = −i
Show Answer

A polynomial with real coefficients includes the conjugate of each square-root zero and each imaginary zero.

x = 6 gives the factor (x − 6).

square root of 6 and −square root of 6 are both zeros.

(x − square root of 6)(x + square root of 6) = x² − 6

−i and i are both zeros.

(x − i)(x + i) = x² + 1

y = (x − 6)(x² − 6)(x² + 1)

Answer: y = (x − 6)(x² − 6)(x² + 1)

4
Write a polynomial of least degree with real coefficients that has these zeros. Leave it in factored form. x = −4 x = square root of 10 x = 3 − 2i
Show Answer

A polynomial with real coefficients includes the conjugate of each square-root zero and each imaginary zero.

x = −4 gives the factor (x + 4).

square root of 10 and −square root of 10 are both zeros.

(x − square root of 10)(x + square root of 10) = x² − 10

3 − 2i and 3 + 2i are both zeros.

(x − (3 − 2i))(x − (3 + 2i))

= (x − 3)² − (2i)²

i² = −1, so (2i)² = −4.

= (x − 3)² + 4

= x² − 6x + 13

y = (x + 4)(x² − 10)(x² − 6x + 13)

Answer: y = (x + 4)(x² − 10)(x² − 6x + 13)

5
Write a polynomial of least degree with real coefficients that has these zeros. Leave it in factored form. x = 1 x = square root of 11 x = 5 + i
Show Answer

A polynomial with real coefficients includes the conjugate of each square-root zero and each imaginary zero.

x = 1 gives the factor (x − 1).

square root of 11 and −square root of 11 are both zeros.

(x − square root of 11)(x + square root of 11) = x² − 11

5 + i and 5 − i are both zeros.

(x − (5 + i))(x − (5 − i))

= (x − 5)² − (i)²

i² = −1, so (i)² = −1.

= (x − 5)² + 1

= x² − 10x + 26

y = (x − 1)(x² − 11)(x² − 10x + 26)

Answer: y = (x − 1)(x² − 11)(x² − 10x + 26)

Level 2

Medium

A zero at 0, a fraction, or a shifted square root.

Example
Write a polynomial of least degree with real coefficients that has these zeros. Leave it in factored form. x = 0 x = −2 x = 3 + square root of 2

A polynomial with real coefficients includes the conjugate of each square-root zero and each imaginary zero.

x = 0 gives the factor x.

x = −2 gives the factor (x + 2).

3 + square root of 2 and 3 − square root of 2 are both zeros.

(x − (3 + square root of 2))(x − (3 − square root of 2))

= (x − 3)² − (square root of 2)²

= (x − 3)² − 2

= x² − 6x + 7

y = x(x + 2)(x² − 6x + 7)

Answer: y = x(x + 2)(x² − 6x + 7)

1
Write a polynomial of least degree with real coefficients that has these zeros. Leave it in factored form. x = 0 x = 4 x = 1 − square root of 3
Show Answer

A polynomial with real coefficients includes the conjugate of each square-root zero and each imaginary zero.

x = 0 gives the factor x.

x = 4 gives the factor (x − 4).

1 + square root of 3 and 1 − square root of 3 are both zeros.

(x − (1 + square root of 3))(x − (1 − square root of 3))

= (x − 1)² − (square root of 3)²

= (x − 1)² − 3

= x² − 2x − 2

y = x(x − 4)(x² − 2x − 2)

Answer: y = x(x − 4)(x² − 2x − 2)

2
Write a polynomial of least degree with real coefficients that has these zeros. Leave it in factored form. x = 0 x = −3 x = 2 + square root of 5
Show Answer

A polynomial with real coefficients includes the conjugate of each square-root zero and each imaginary zero.

x = 0 gives the factor x.

x = −3 gives the factor (x + 3).

2 + square root of 5 and 2 − square root of 5 are both zeros.

(x − (2 + square root of 5))(x − (2 − square root of 5))

= (x − 2)² − (square root of 5)²

= (x − 2)² − 5

= x² − 4x − 1

y = x(x + 3)(x² − 4x − 1)

Answer: y = x(x + 3)(x² − 4x − 1)

3
Write a polynomial of least degree with real coefficients that has these zeros. Leave it in factored form. x = −12 x = 6 x = isquare root of 3
Show Answer

A polynomial with real coefficients includes the conjugate of each square-root zero and each imaginary zero.

x = −12 means 2x + 1 = 0.

The factor is (2x + 1).

x = 6 gives the factor (x − 6).

isquare root of 3 and −isquare root of 3 are both zeros.

(x − isquare root of 3)(x + isquare root of 3) = x² + 3

y = (2x + 1)(x − 6)(x² + 3)

Answer: y = (2x + 1)(x − 6)(x² + 3)

4
Write a polynomial of least degree with real coefficients that has these zeros. Leave it in factored form. x = 23 x = −5 x = 1 − i
Show Answer

A polynomial with real coefficients includes the conjugate of each square-root zero and each imaginary zero.

x = 23 means 3x − 2 = 0.

The factor is (3x − 2).

x = −5 gives the factor (x + 5).

1 − i and 1 + i are both zeros.

(x − (1 − i))(x − (1 + i))

= (x − 1)² − (i)²

i² = −1, so (i)² = −1.

= (x − 1)² + 1

= x² − 2x + 2

y = (3x − 2)(x + 5)(x² − 2x + 2)

Answer: y = (3x − 2)(x + 5)(x² − 2x + 2)

5
Write a polynomial of least degree with real coefficients that has these zeros. Leave it in factored form. x = 0 x = square root of 13 x = 4 − i
Show Answer

A polynomial with real coefficients includes the conjugate of each square-root zero and each imaginary zero.

x = 0 gives the factor x.

square root of 13 and −square root of 13 are both zeros.

(x − square root of 13)(x + square root of 13) = x² − 13

4 − i and 4 + i are both zeros.

(x − (4 − i))(x − (4 + i))

= (x − 4)² − (i)²

i² = −1, so (i)² = −1.

= (x − 4)² + 1

= x² − 8x + 17

y = x(x² − 13)(x² − 8x + 17)

Answer: y = x(x² − 13)(x² − 8x + 17)

Level 3

Hard

A fraction, an imaginary pair, and a square-root pair together.

Example
Write a polynomial of least degree with real coefficients that has these zeros. Leave it in factored form. x = 4 x = −2i x = 1 + square root of 5

A polynomial with real coefficients includes the conjugate of each square-root zero and each imaginary zero.

x = 4 gives the factor (x − 4).

−2i and 2i are both zeros.

(x − 2i)(x + 2i) = x² + 4

1 + square root of 5 and 1 − square root of 5 are both zeros.

(x − (1 + square root of 5))(x − (1 − square root of 5))

= (x − 1)² − (square root of 5)²

= (x − 1)² − 5

= x² − 2x − 4

y = (x − 4)(x² + 4)(x² − 2x − 4)

Answer: y = (x − 4)(x² + 4)(x² − 2x − 4)

1
Write a polynomial of least degree with real coefficients that has these zeros. Leave it in factored form. x = −32 x = 2i x = 4 + square root of 3
Show Answer

A polynomial with real coefficients includes the conjugate of each square-root zero and each imaginary zero.

x = −32 means 2x + 3 = 0.

The factor is (2x + 3).

2i and −2i are both zeros.

(x − 2i)(x + 2i) = x² + 4

4 + square root of 3 and 4 − square root of 3 are both zeros.

(x − (4 + square root of 3))(x − (4 − square root of 3))

= (x − 4)² − (square root of 3)²

= (x − 4)² − 3

= x² − 8x + 13

y = (2x + 3)(x² + 4)(x² − 8x + 13)

Answer: y = (2x + 3)(x² + 4)(x² − 8x + 13)

2
Write a polynomial of least degree with real coefficients that has these zeros. Leave it in factored form. x = 52 x = −3i x = square root of 6
Show Answer

A polynomial with real coefficients includes the conjugate of each square-root zero and each imaginary zero.

x = 52 means 2x − 5 = 0.

The factor is (2x − 5).

−3i and 3i are both zeros.

(x − 3i)(x + 3i) = x² + 9

square root of 6 and −square root of 6 are both zeros.

(x − square root of 6)(x + square root of 6) = x² − 6

y = (2x − 5)(x² + 9)(x² − 6)

Answer: y = (2x − 5)(x² + 9)(x² − 6)

3
Write a polynomial of least degree with real coefficients that has these zeros. Leave it in factored form. x = −43 x = isquare root of 5 x = 2 − square root of 7
Show Answer

A polynomial with real coefficients includes the conjugate of each square-root zero and each imaginary zero.

x = −43 means 3x + 4 = 0.

The factor is (3x + 4).

isquare root of 5 and −isquare root of 5 are both zeros.

(x − isquare root of 5)(x + isquare root of 5) = x² + 5

2 + square root of 7 and 2 − square root of 7 are both zeros.

(x − (2 + square root of 7))(x − (2 − square root of 7))

= (x − 2)² − (square root of 7)²

= (x − 2)² − 7

= x² − 4x − 3

y = (3x + 4)(x² + 5)(x² − 4x − 3)

Answer: y = (3x + 4)(x² + 5)(x² − 4x − 3)

4
Write a polynomial of least degree with real coefficients that has these zeros. Leave it in factored form. x = 0 x = −72 x = 3 + 2i
Show Answer

A polynomial with real coefficients includes the conjugate of each square-root zero and each imaginary zero.

x = 0 gives the factor x.

x = −72 means 2x + 7 = 0.

The factor is (2x + 7).

3 + 2i and 3 − 2i are both zeros.

(x − (3 + 2i))(x − (3 − 2i))

= (x − 3)² − (2i)²

i² = −1, so (2i)² = −4.

= (x − 3)² + 4

= x² − 6x + 13

y = x(2x + 7)(x² − 6x + 13)

Answer: y = x(2x + 7)(x² − 6x + 13)

5
Write a polynomial of least degree with real coefficients that has these zeros. Leave it in factored form. x = 14 x = square root of 2 x = 5 − i
Show Answer

A polynomial with real coefficients includes the conjugate of each square-root zero and each imaginary zero.

x = 14 means 4x − 1 = 0.

The factor is (4x − 1).

square root of 2 and −square root of 2 are both zeros.

(x − square root of 2)(x + square root of 2) = x² − 2

5 − i and 5 + i are both zeros.

(x − (5 − i))(x − (5 + i))

= (x − 5)² − (i)²

i² = −1, so (i)² = −1.

= (x − 5)² + 1

= x² − 10x + 26

y = (4x − 1)(x² − 2)(x² − 10x + 26)

Answer: y = (4x − 1)(x² − 2)(x² − 10x + 26)

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