Polynomial from Zeros | Math Coach Amy

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Key Ideas Square Root Imaginary x = 0 2 + square root of 3 3 + i Mix

Subtract each zero from both sides.

Include the conjugate, then multiply that pair.

What to Remember

Subtract from both sides

Write each zero. Subtract that number from both sides.

For x = 0, the factor (x − 0) is (x).

Multiply the conjugate pair

square root of 5 comes with −square root of 5. 2i comes with −2i.

2 + square root of 3 comes with 2 − square root of 3. 3 + i comes with 3 − i.

Multiply those two factors. The quadratic has real numbers only.

Easy

Integer and a Square Root

One integer zero and one square-root zero.

Example

Write a polynomial of least degree with real coefficients that has these zeros. x = 2 and x = square root of 5

x = 2

x = square root of 5

x = −square root of 5

−2−2

−square root of 5−square root of 5

+ square root of 5+ square root of 5

x − 2 = 0

x − square root of 5 = 0

x + square root of 5 = 0

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

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

x² + xsquare root of 5 − xsquare root of 5 − square root of 5 · square root of 5

x² − 5

y = (x − 2)(x² − 5)

Answer: y = (x − 2)(x² − 5)

Write a polynomial of least degree with real coefficients that has these zeros. x = 4 and x = square root of 3

Show Answer

x = 4

x = square root of 3

x = −square root of 3

−4−4

−square root of 3−square root of 3

+ square root of 3+ square root of 3

x − 4 = 0

x − square root of 3 = 0

x + square root of 3 = 0

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

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

x² + xsquare root of 3 − xsquare root of 3 − square root of 3 · square root of 3

x² − 3

y = (x − 4)(x² − 3)

Answer: y = (x − 4)(x² − 3)

Write a polynomial of least degree with real coefficients that has these zeros. x = −1 and x = square root of 7

Show Answer

x = −1

x = square root of 7

x = −square root of 7

+ 1+ 1

−square root of 7−square root of 7

+ square root of 7+ square root of 7

x + 1 = 0

x − square root of 7 = 0

x + square root of 7 = 0

y = (x + 1)(x − square root of 7)(x + square root of 7)

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

x² + xsquare root of 7 − xsquare root of 7 − square root of 7 · square root of 7

x² − 7

y = (x + 1)(x² − 7)

Answer: y = (x + 1)(x² − 7)

Write a polynomial of least degree with real coefficients that has these zeros. x = 6 and x = square root of 2

Show Answer

x = 6

x = square root of 2

x = −square root of 2

−6−6

−square root of 2−square root of 2

+ square root of 2+ square root of 2

x − 6 = 0

x − square root of 2 = 0

x + square root of 2 = 0

y = (x − 6)(x − square root of 2)(x + square root of 2)

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

x² + xsquare root of 2 − xsquare root of 2 − square root of 2 · square root of 2

x² − 2

y = (x − 6)(x² − 2)

Answer: y = (x − 6)(x² − 2)

Write a polynomial of least degree with real coefficients that has these zeros. x = −5 and x = square root of 6

Show Answer

x = −5

x = square root of 6

x = −square root of 6

+ 5+ 5

−square root of 6−square root of 6

+ square root of 6+ square root of 6

x + 5 = 0

x − square root of 6 = 0

x + square root of 6 = 0

y = (x + 5)(x − square root of 6)(x + square root of 6)

Multiply (x − square root of 6)(x + square root of 6)

x² + xsquare root of 6 − xsquare root of 6 − square root of 6 · square root of 6

x² − 6

y = (x + 5)(x² − 6)

Answer: y = (x + 5)(x² − 6)

Write a polynomial of least degree with real coefficients that has these zeros. x = 8 and x = square root of 10

Show Answer

x = 8

x = square root of 10

x = −square root of 10

−8−8

−square root of 10−square root of 10

+ square root of 10+ square root of 10

x − 8 = 0

x − square root of 10 = 0

x + square root of 10 = 0

y = (x − 8)(x − square root of 10)(x + square root of 10)

Multiply (x − square root of 10)(x + square root of 10)

x² + xsquare root of 10 − xsquare root of 10 − square root of 10 · square root of 10

x² − 10

y = (x − 8)(x² − 10)

Answer: y = (x − 8)(x² − 10)

Easy

Integer and an Imaginary Number

One integer zero and one imaginary zero.

Example

Write a polynomial of least degree with real coefficients that has these zeros. x = 3 and x = 2i

x = 3

x = 2i

x = −2i

−3−3

−2i−2i

+ 2i+ 2i

x − 3 = 0

x − 2i = 0

x + 2i = 0

y = (x − 3)(x − 2i)(x + 2i)

Multiply (x − 2i)(x + 2i)

x² + x(2i) − x(2i) − (2i) · (2i)

x² − 4i²

i² = −1

x² − 4(−1)

x² + 4

y = (x − 3)(x² + 4)

Answer: y = (x − 3)(x² + 4)

Write a polynomial of least degree with real coefficients that has these zeros. x = 5 and x = 4i

Show Answer

x = 5

x = 4i

x = −4i

−5−5

−4i−4i

+ 4i+ 4i

x − 5 = 0

x − 4i = 0

x + 4i = 0

y = (x − 5)(x − 4i)(x + 4i)

Multiply (x − 4i)(x + 4i)

x² + x(4i) − x(4i) − (4i) · (4i)

x² − 16i²

i² = −1

x² − 16(−1)

x² + 16

y = (x − 5)(x² + 16)

Answer: y = (x − 5)(x² + 16)

Write a polynomial of least degree with real coefficients that has these zeros. x = −2 and x = i

Show Answer

x = −2

x = i

x = −i

+ 2+ 2

−i−i

+ i+ i

x + 2 = 0

x − i = 0

x + i = 0

y = (x + 2)(x − i)(x + i)

Multiply (x − i)(x + i)

x² + x(i) − x(i) − (i) · (i)

x² − i²

i² = −1

x² − (−1)

x² + 1

y = (x + 2)(x² + 1)

Answer: y = (x + 2)(x² + 1)

Write a polynomial of least degree with real coefficients that has these zeros. x = 7 and x = 3i

Show Answer

x = 7

x = 3i

x = −3i

−7−7

−3i−3i

+ 3i+ 3i

x − 7 = 0

x − 3i = 0

x + 3i = 0

y = (x − 7)(x − 3i)(x + 3i)

Multiply (x − 3i)(x + 3i)

x² + x(3i) − x(3i) − (3i) · (3i)

x² − 9i²

i² = −1

x² − 9(−1)

x² + 9

y = (x − 7)(x² + 9)

Answer: y = (x − 7)(x² + 9)

Write a polynomial of least degree with real coefficients that has these zeros. x = −6 and x = 5i

Show Answer

x = −6

x = 5i

x = −5i

+ 6+ 6

−5i−5i

+ 5i+ 5i

x + 6 = 0

x − 5i = 0

x + 5i = 0

y = (x + 6)(x − 5i)(x + 5i)

Multiply (x − 5i)(x + 5i)

x² + x(5i) − x(5i) − (5i) · (5i)

x² − 25i²

i² = −1

x² − 25(−1)

x² + 25

y = (x + 6)(x² + 25)

Answer: y = (x + 6)(x² + 25)

Write a polynomial of least degree with real coefficients that has these zeros. x = 1 and x = 6i

Show Answer

x = 1

x = 6i

x = −6i

−1−1

−6i−6i

+ 6i+ 6i

x − 1 = 0

x − 6i = 0

x + 6i = 0

y = (x − 1)(x − 6i)(x + 6i)

Multiply (x − 6i)(x + 6i)

x² + x(6i) − x(6i) − (6i) · (6i)

x² − 36i²

i² = −1

x² − 36(−1)

x² + 36

y = (x − 1)(x² + 36)

Answer: y = (x − 1)(x² + 36)

Easy

Integers and Zero

Two integer zeros and x = 0.

Example

Write a polynomial of least degree with real coefficients that has these zeros. x = 4, x = −3, and x = 0

x = 4

x = −3

x = 0

−4−4

+ 3+ 3

−0−0

x − 4 = 0

x + 3 = 0

x − 0 = 0

y = (x − 4)(x + 3)(x − 0)

(x − 0) = (x)

y = (x − 4)(x + 3)(x)

Answer: y = (x − 4)(x + 3)(x)

Write a polynomial of least degree with real coefficients that has these zeros. x = 2, x = 5, and x = 0

Show Answer

x = 2

x = 5

x = 0

−2−2

−5−5

−0−0

x − 2 = 0

x − 5 = 0

x − 0 = 0

y = (x − 2)(x − 5)(x − 0)

(x − 0) = (x)

y = (x − 2)(x − 5)(x)

Answer: y = (x − 2)(x − 5)(x)

Write a polynomial of least degree with real coefficients that has these zeros. x = −1, x = 6, and x = 0

Show Answer

x = −1

x = 6

x = 0

+ 1+ 1

−6−6

−0−0

x + 1 = 0

x − 6 = 0

x − 0 = 0

y = (x + 1)(x − 6)(x − 0)

(x − 0) = (x)

y = (x + 1)(x − 6)(x)

Answer: y = (x + 1)(x − 6)(x)

Write a polynomial of least degree with real coefficients that has these zeros. x = −4, x = −2, and x = 0

Show Answer

x = −4

x = −2

x = 0

+ 4+ 4

+ 2+ 2

−0−0

x + 4 = 0

x + 2 = 0

x − 0 = 0

y = (x + 4)(x + 2)(x − 0)

(x − 0) = (x)

y = (x + 4)(x + 2)(x)

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

Write a polynomial of least degree with real coefficients that has these zeros. x = 7, x = −8, and x = 0

Show Answer

x = 7

x = −8

x = 0

−7−7

+ 8+ 8

−0−0

x − 7 = 0

x + 8 = 0

x − 0 = 0

y = (x − 7)(x + 8)(x − 0)

(x − 0) = (x)

y = (x − 7)(x + 8)(x)

Answer: y = (x − 7)(x + 8)(x)

Write a polynomial of least degree with real coefficients that has these zeros. x = 9, x = 3, and x = 0

Show Answer

x = 9

x = 3

x = 0

−9−9

−3−3

−0−0

x − 9 = 0

x − 3 = 0

x − 0 = 0

y = (x − 9)(x − 3)(x − 0)

(x − 0) = (x)

y = (x − 9)(x − 3)(x)

Answer: y = (x − 9)(x − 3)(x)

Medium

Integer and a Shifted Square Root

One integer zero and one zero such as 2 + square root of 3.

Example

Write a polynomial of least degree with real coefficients that has these zeros. x = 5 and x = 2 + square root of 3

x = 5

x = 2 + square root of 3

x = 2 − square root of 3

−5−5

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

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

x − 5 = 0

x − (2 + square root of 3) = 0

x − (2 − square root of 3) = 0

y = (x − 5)(x − (2 + square root of 3))(x − (2 − square root of 3))

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

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

(x − 2)² − (square root of 3)²

(x − 2)² − 3

x² − 4x + 4 − 3

x² − 4x + 1

y = (x − 5)(x² − 4x + 1)

Answer: y = (x − 5)(x² − 4x + 1)

Write a polynomial of least degree with real coefficients that has these zeros. x = −1 and x = 3 + square root of 2

Show Answer

x = −1

x = 3 + square root of 2

x = 3 − square root of 2

+ 1+ 1

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

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

x + 1 = 0

x − (3 + square root of 2) = 0

x − (3 − square root of 2) = 0

y = (x + 1)(x − (3 + square root of 2))(x − (3 − square root of 2))

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

(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 + 9 − 2

x² − 6x + 7

y = (x + 1)(x² − 6x + 7)

Answer: y = (x + 1)(x² − 6x + 7)

Write a polynomial of least degree with real coefficients that has these zeros. x = 6 and x = 1 + square root of 5

Show Answer

x = 6

x = 1 + square root of 5

x = 1 − square root of 5

−6−6

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

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

x − 6 = 0

x − (1 + square root of 5) = 0

x − (1 − square root of 5) = 0

y = (x − 6)(x − (1 + square root of 5))(x − (1 − square root of 5))

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

(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 + 1 − 5

x² − 2x − 4

y = (x − 6)(x² − 2x − 4)

Answer: y = (x − 6)(x² − 2x − 4)

Write a polynomial of least degree with real coefficients that has these zeros. x = −4 and x = 4 + square root of 7

Show Answer

x = −4

x = 4 + square root of 7

x = 4 − square root of 7

+ 4+ 4

−(4 + square root of 7)−(4 + square root of 7)

−(4 − square root of 7)−(4 − square root of 7)

x + 4 = 0

x − (4 + square root of 7) = 0

x − (4 − square root of 7) = 0

y = (x + 4)(x − (4 + square root of 7))(x − (4 − square root of 7))

Multiply (x − (4 + square root of 7))(x − (4 − square root of 7))

(x − 4 − square root of 7)(x − 4 + square root of 7)

(x − 4)² − (square root of 7)²

(x − 4)² − 7

x² − 8x + 16 − 7

x² − 8x + 9

y = (x + 4)(x² − 8x + 9)

Answer: y = (x + 4)(x² − 8x + 9)

Write a polynomial of least degree with real coefficients that has these zeros. x = 8 and x = 2 + square root of 11

Show Answer

x = 8

x = 2 + square root of 11

x = 2 − square root of 11

−8−8

−(2 + square root of 11)−(2 + square root of 11)

−(2 − square root of 11)−(2 − square root of 11)

x − 8 = 0

x − (2 + square root of 11) = 0

x − (2 − square root of 11) = 0

y = (x − 8)(x − (2 + square root of 11))(x − (2 − square root of 11))

Multiply (x − (2 + square root of 11))(x − (2 − square root of 11))

(x − 2 − square root of 11)(x − 2 + square root of 11)

(x − 2)² − (square root of 11)²

(x − 2)² − 11

x² − 4x + 4 − 11

x² − 4x − 7

y = (x − 8)(x² − 4x − 7)

Answer: y = (x − 8)(x² − 4x − 7)

Write a polynomial of least degree with real coefficients that has these zeros. x = 3 and x = 5 + square root of 6

Show Answer

x = 3

x = 5 + square root of 6

x = 5 − square root of 6

−3−3

−(5 + square root of 6)−(5 + square root of 6)

−(5 − square root of 6)−(5 − square root of 6)

x − 3 = 0

x − (5 + square root of 6) = 0

x − (5 − square root of 6) = 0

y = (x − 3)(x − (5 + square root of 6))(x − (5 − square root of 6))

Multiply (x − (5 + square root of 6))(x − (5 − square root of 6))

(x − 5 − square root of 6)(x − 5 + square root of 6)

(x − 5)² − (square root of 6)²

(x − 5)² − 6

x² − 10x + 25 − 6

x² − 10x + 19

y = (x − 3)(x² − 10x + 19)

Answer: y = (x − 3)(x² − 10x + 19)

Medium

Integer and a + bi

One integer zero and one zero such as 3 + i.

Example

Write a polynomial of least degree with real coefficients that has these zeros. x = 4 and x = 3 + i

x = 4

x = 3 + i

x = 3 − i

−4−4

−(3 + i)−(3 + i)

−(3 − i)−(3 − i)

x − 4 = 0

x − (3 + i) = 0

x − (3 − i) = 0

y = (x − 4)(x − (3 + i))(x − (3 − i))

Multiply (x − (3 + i))(x − (3 − i))

(x − 3 − i)(x − 3 + i)

(x − 3)² − i²

i² = −1

(x − 3)² − (−1)

(x − 3)² + 1

x² − 6x + 9 + 1

x² − 6x + 10

y = (x − 4)(x² − 6x + 10)

Answer: y = (x − 4)(x² − 6x + 10)

Write a polynomial of least degree with real coefficients that has these zeros. x = 1 and x = 2 + i

Show Answer

x = 1

x = 2 + i

x = 2 − i

−1−1

−(2 + i)−(2 + i)

−(2 − i)−(2 − i)

x − 1 = 0

x − (2 + i) = 0

x − (2 − i) = 0

y = (x − 1)(x − (2 + i))(x − (2 − i))

Multiply (x − (2 + i))(x − (2 − i))

(x − 2 − i)(x − 2 + i)

(x − 2)² − i²

i² = −1

(x − 2)² − (−1)

(x − 2)² + 1

x² − 4x + 4 + 1

x² − 4x + 5

y = (x − 1)(x² − 4x + 5)

Answer: y = (x − 1)(x² − 4x + 5)

Write a polynomial of least degree with real coefficients that has these zeros. x = −2 and x = 4 + 2i

Show Answer

x = −2

x = 4 + 2i

x = 4 − 2i

+ 2+ 2

−(4 + 2i)−(4 + 2i)

−(4 − 2i)−(4 − 2i)

x + 2 = 0

x − (4 + 2i) = 0

x − (4 − 2i) = 0

y = (x + 2)(x − (4 + 2i))(x − (4 − 2i))

Multiply (x − (4 + 2i))(x − (4 − 2i))

(x − 4 − 2i)(x − 4 + 2i)

(x − 4)² − (2i)²

(x − 4)² − 4i²

i² = −1

(x − 4)² − 4(−1)

(x − 4)² + 4

x² − 8x + 16 + 4

x² − 8x + 20

y = (x + 2)(x² − 8x + 20)

Answer: y = (x + 2)(x² − 8x + 20)

Write a polynomial of least degree with real coefficients that has these zeros. x = 6 and x = 1 + 3i

Show Answer

x = 6

x = 1 + 3i

x = 1 − 3i

−6−6

−(1 + 3i)−(1 + 3i)

−(1 − 3i)−(1 − 3i)

x − 6 = 0

x − (1 + 3i) = 0

x − (1 − 3i) = 0

y = (x − 6)(x − (1 + 3i))(x − (1 − 3i))

Multiply (x − (1 + 3i))(x − (1 − 3i))

(x − 1 − 3i)(x − 1 + 3i)

(x − 1)² − (3i)²

(x − 1)² − 9i²

i² = −1

(x − 1)² − 9(−1)

(x − 1)² + 9

x² − 2x + 1 + 9

x² − 2x + 10

y = (x − 6)(x² − 2x + 10)

Answer: y = (x − 6)(x² − 2x + 10)

Write a polynomial of least degree with real coefficients that has these zeros. x = −4 and x = 2 + 4i

Show Answer

x = −4

x = 2 + 4i

x = 2 − 4i

+ 4+ 4

−(2 + 4i)−(2 + 4i)

−(2 − 4i)−(2 − 4i)

x + 4 = 0

x − (2 + 4i) = 0

x − (2 − 4i) = 0

y = (x + 4)(x − (2 + 4i))(x − (2 − 4i))

Multiply (x − (2 + 4i))(x − (2 − 4i))

(x − 2 − 4i)(x − 2 + 4i)

(x − 2)² − (4i)²

(x − 2)² − 16i²

i² = −1

(x − 2)² − 16(−1)

(x − 2)² + 16

x² − 4x + 4 + 16

x² − 4x + 20

y = (x + 4)(x² − 4x + 20)

Answer: y = (x + 4)(x² − 4x + 20)

Write a polynomial of least degree with real coefficients that has these zeros. x = 7 and x = 5 + 2i

Show Answer

x = 7

x = 5 + 2i

x = 5 − 2i

−7−7

−(5 + 2i)−(5 + 2i)

−(5 − 2i)−(5 − 2i)

x − 7 = 0

x − (5 + 2i) = 0

x − (5 − 2i) = 0

y = (x − 7)(x − (5 + 2i))(x − (5 − 2i))

Multiply (x − (5 + 2i))(x − (5 − 2i))

(x − 5 − 2i)(x − 5 + 2i)

(x − 5)² − (2i)²

(x − 5)² − 4i²

i² = −1

(x − 5)² − 4(−1)

(x − 5)² + 4

x² − 10x + 25 + 4

x² − 10x + 29

y = (x − 7)(x² − 10x + 29)

Answer: y = (x − 7)(x² − 10x + 29)

Hard

Mixed Zeros

Integer, square-root, imaginary, and zero solutions together.

Example

Write a polynomial of least degree with real coefficients that has these zeros. x = 2, x = square root of 3, and x = 4i

x = 2

x = square root of 3

x = −square root of 3

x = 4i

x = −4i

−2−2

−square root of 3−square root of 3

+ square root of 3+ square root of 3

−4i−4i

+ 4i+ 4i

x − 2 = 0

x − square root of 3 = 0

x + square root of 3 = 0

x − 4i = 0

x + 4i = 0

y = (x − 2)(x − square root of 3)(x + square root of 3)(x − 4i)(x + 4i)

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

x² + xsquare root of 3 − xsquare root of 3 − square root of 3 · square root of 3

x² − 3

Multiply (x − 4i)(x + 4i)

x² + x(4i) − x(4i) − (4i) · (4i)

x² − 16i²

i² = −1

x² − 16(−1)

x² + 16

y = (x − 2)(x² − 3)(x² + 16)

Answer: y = (x − 2)(x² − 3)(x² + 16)

Write a polynomial of least degree with real coefficients that has these zeros. x = −5, x = 0, and x = 2 + square root of 5

Show Answer

x = −5

x = 0

x = 2 + square root of 5

x = 2 − square root of 5

+ 5+ 5

−0−0

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

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

x + 5 = 0

x − 0 = 0

x − (2 + square root of 5) = 0

x − (2 − square root of 5) = 0

y = (x + 5)(x − 0)(x − (2 + square root of 5))(x − (2 − square root of 5))

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

(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 + 4 − 5

x² − 4x − 1

(x − 0) = (x)

y = (x + 5)(x)(x² − 4x − 1)

Answer: y = (x + 5)(x)(x² − 4x − 1)

Write a polynomial of least degree with real coefficients that has these zeros. x = 10, x = 4 + i, and x = square root of 7

Show Answer

x = 10

x = 4 + i

x = 4 − i

x = square root of 7

x = −square root of 7

−10−10

−(4 + i)−(4 + i)

−(4 − i)−(4 − i)

−square root of 7−square root of 7

+ square root of 7+ square root of 7

x − 10 = 0

x − (4 + i) = 0

x − (4 − i) = 0

x − square root of 7 = 0

x + square root of 7 = 0

y = (x − 10)(x − (4 + i))(x − (4 − i))(x − square root of 7)(x + square root of 7)

Multiply (x − (4 + i))(x − (4 − i))

(x − 4 − i)(x − 4 + i)

(x − 4)² − i²

i² = −1

(x − 4)² − (−1)

(x − 4)² + 1

x² − 8x + 16 + 1

x² − 8x + 17

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

x² + xsquare root of 7 − xsquare root of 7 − square root of 7 · square root of 7

x² − 7

y = (x − 10)(x² − 8x + 17)(x² − 7)

Answer: y = (x − 10)(x² − 8x + 17)(x² − 7)

Write a polynomial of least degree with real coefficients that has these zeros. x = −2, x = 7, x = 5 + square root of 2, and x = 3i

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x = −2

x = 7

x = 5 + square root of 2

x = 5 − square root of 2

x = 3i

x = −3i

+ 2+ 2

−7−7

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

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

−3i−3i

+ 3i+ 3i

x + 2 = 0

x − 7 = 0

x − (5 + square root of 2) = 0

x − (5 − square root of 2) = 0

x − 3i = 0

x + 3i = 0

y = (x + 2)(x − 7)(x − (5 + square root of 2))(x − (5 − square root of 2))(x − 3i)(x + 3i)

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

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

(x − 5)² − (square root of 2)²

(x − 5)² − 2

x² − 10x + 25 − 2

x² − 10x + 23

Multiply (x − 3i)(x + 3i)

x² + x(3i) − x(3i) − (3i) · (3i)

x² − 9i²

i² = −1

x² − 9(−1)

x² + 9

y = (x + 2)(x − 7)(x² − 10x + 23)(x² + 9)

Answer: y = (x + 2)(x − 7)(x² − 10x + 23)(x² + 9)

Write a polynomial of least degree with real coefficients that has these zeros. x = 0, x = 1 + 2i, and x = square root of 10

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x = 0

x = 1 + 2i

x = 1 − 2i

x = square root of 10

x = −square root of 10

−0−0

−(1 + 2i)−(1 + 2i)

−(1 − 2i)−(1 − 2i)

−square root of 10−square root of 10

+ square root of 10+ square root of 10

x − 0 = 0

x − (1 + 2i) = 0

x − (1 − 2i) = 0

x − square root of 10 = 0

x + square root of 10 = 0

y = (x − 0)(x − (1 + 2i))(x − (1 − 2i))(x − square root of 10)(x + square root of 10)

Multiply (x − (1 + 2i))(x − (1 − 2i))

(x − 1 − 2i)(x − 1 + 2i)

(x − 1)² − (2i)²

(x − 1)² − 4i²

i² = −1

(x − 1)² − 4(−1)

(x − 1)² + 4

x² − 2x + 1 + 4

x² − 2x + 5

Multiply (x − square root of 10)(x + square root of 10)

x² + xsquare root of 10 − xsquare root of 10 − square root of 10 · square root of 10

x² − 10

(x − 0) = (x)

y = (x)(x² − 2x + 5)(x² − 10)

Answer: y = (x)(x² − 2x + 5)(x² − 10)

Write a polynomial of least degree with real coefficients that has these zeros. x = 9, x = 4 + square root of 3, and x = 6 + 2i

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x = 9

x = 4 + square root of 3

x = 4 − square root of 3

x = 6 + 2i

x = 6 − 2i

−9−9

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

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

−(6 + 2i)−(6 + 2i)

−(6 − 2i)−(6 − 2i)

x − 9 = 0

x − (4 + square root of 3) = 0

x − (4 − square root of 3) = 0

x − (6 + 2i) = 0

x − (6 − 2i) = 0

y = (x − 9)(x − (4 + square root of 3))(x − (4 − square root of 3))(x − (6 + 2i))(x − (6 − 2i))

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

(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 + 16 − 3

x² − 8x + 13

Multiply (x − (6 + 2i))(x − (6 − 2i))

(x − 6 − 2i)(x − 6 + 2i)

(x − 6)² − (2i)²

(x − 6)² − 4i²

i² = −1

(x − 6)² − 4(−1)

(x − 6)² + 4

x² − 12x + 36 + 4

x² − 12x + 40

y = (x − 9)(x² − 8x + 13)(x² − 12x + 40)

Answer: y = (x − 9)(x² − 8x + 13)(x² − 12x + 40)

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