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This printable supports Common Core Mathematics Standard HSN-CN.C.8

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# Polynomial Identities and Complex Numbers (Grades 11-12)

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## Polynomial Identities and Complex Numbers

1.
Factor. $x^2+16$
1. $sqrt16i$
2. $(x-4i)(x+4i)$
3. $+-sqrt4$
4. $(x+2i)(x-2i)$
2.
Factor. $x^2+12$
1. $2sqrt3i$
2. $(x-4i)(x+4i)$
3. $+-sqrt12$
4. $(x+2sqrt3i)(x-2sqrt3i)$
3.
Factor. $x^2+64$
1. $(x+8i)(x-8i)$
2. $(x-8)(x+8)$
3. $+-sqrt64$
4. $(x+64i)(x-64i)$
4.
Which of the following is the factored form of $x^3 + 8 ?$
1. $(x+2)(x-1+sqrt(3) \ i)(x-1-sqrt(3) \ i)$
2. $(x+2)(x-2)(x+2)$
3. $(x+2)(x-2i)(x+2i)$
4. $(x+2)(x+2+2sqrt(3) \ i)(x-2-2sqrt(3) \ i)$
5.
Factor $x^4-1$.
1. $(x^2+1)(x+1)(x+i)$
2. $(x+1)(x-1)(x+1+i)(x-1-i)$
3. $(x+1)(x-1)(x+i)(x-i)$
4. $(x+1)(x-1)(x+sqrt(2)i)(x-sqrt(2)i)$
6.
If $x - 3/2 + 3/2sqrt(3) \ i$ is a factor of $x^3 + 27$, which of the following are also factors of this polynomial?
1. $x - 3$
2. $x + 3/2 - 3/2 sqrt(3)$
3. $x - 3/2 - 3/2 sqrt(3) \ i$
4. $x + 3/2 - 3/2 sqrt(3) \ i$
7.
If $x+2(1-sqrt(3)i)$ is a factor of $x^3-64$, which of the following are also factors? Choose all that apply.
1. $x-4$
2. $x+2(1+sqrt(3)i)$
3. $x-2(1-sqrt(3)i)$
4. $x-2(1+sqrt(3)i)$
8.
Which of the following are equivalent to $x^4 - 16 ?$ Choose all that apply.
1. $(x+2)(x^3-2i \ x^2 - 4x + 8i)$
2. $(x+2)(x-2)(x+2i)(x-2i)$
3. $(x+2)(x-2)(x^2+4)$
4. $(x+2i)(x-2i)(x^2-4)$
9.
Which of the following are equivalent to $4x^4 - 81 ?$ Choose all that apply.
1. $(sqrt(2)x - 3)(sqrt(2)x + 3)(sqrt(2)x - 3i)(sqrt(2)x + 3i)$
2. $(root[4](4)x - 3)^2(root[4](4)x + 3)^2$
3. $(2x^2 - 9)(2x^2 + 9)$
4. $(1/2 x + 3)(1/2 x -3) (2x^2 + 9)$
10.
For the polynomial $P(x) = (ax)^3 + b^3$, $a,b in RR$, and where $a,b$ are positive, if $1 + sqrt(3)i$ is a root, and in fully factored form there is a leading constant of 8, what are the coefficients $a,b ?$ (Assume that the coefficient of 8 has come from factoring out a 2 from each of the three factors of $P(x)$).
1. $a=1, b=2$
2. $a=1, b=4$
3. $a=2, b=4$
4. Not enough information to solve.
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