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Back to Exercise: Extend polynomial identities to complex numbers

Exercises: Extend Polynomial Identities to Complex Numbers

Factor completely over the complex numbers. Verify by expanding when asked.

Grade 9·20 problems·~40 min·Common Core Math - HS Number and Quantity·standard·hsn-cn-c-8
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A

Warm-Up: Review What You Know

1

Factor x2−9x^2 - 9 over the real numbers.

2

Compute (x+2i)(x−2i)(x + 2i)(x - 2i). Simplify using i2=−1i^2 = -1.

3

The quadratic x2−4x+5=0x^2 - 4x + 5 = 0 has roots 2+i2 + i and 2−i2 - i (from CN.C.7).

By the root-to-factor connection: x2−4x+5=(x−r1)(x−r2)x^2 - 4x + 5 = (x - r_1)(x - r_2).

Verify: expand (x−(2+i))(x−(2−i))(x - (2+i))(x - (2-i)) and find the constant term. Enter the constant term.

B

Fluency Practice

1

Factor x2+9x^2 + 9 over the complex numbers.

2

The canonical example: factor x2+4x^2 + 4 over the complex numbers.

3

Factor x2+5x^2 + 5 over the complex numbers.

4

The quadratic x2+2x+5=0x^2 + 2x + 5 = 0 has roots −1+2i-1 + 2i and −1−2i-1 - 2i.

Write the factored form: $x^2 + 2x + 5 = (x - $   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   $)(x - $   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   )$.

Enter the two roots (the numbers being subtracted), separated.

first root (with sign):
second root (with sign):
5

Factor 4x2+254x^2 + 25 over the complex numbers.

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