Extending Polynomial Identities to Complex Numbers - Algebra 2
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Rewrite $x^2+36$ as a product of two complex conjugate binomials.
Rewrite $x^2+36$ as a product of two complex conjugate binomials.
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$(x+6i)(x-6i)$. $x^2 + 36 = x^2 + (6i)^2$ factors as conjugate pair.
$(x+6i)(x-6i)$. $x^2 + 36 = x^2 + (6i)^2$ factors as conjugate pair.
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What is $i^{27}$?
What is $i^{27}$?
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$i^{27}=-i$. $i^{27} = i^{24} \cdot i^3 = 1 \cdot (-i) = -i$
$i^{27}=-i$. $i^{27} = i^{24} \cdot i^3 = 1 \cdot (-i) = -i$
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Factor $x^2+6x+13$ over the complex numbers.
Factor $x^2+6x+13$ over the complex numbers.
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$(x+3+2i)(x+3-2i)$. Complete the square: $(x+3)^2 - 9 + 13 = (x+3)^2 + 4$.
$(x+3+2i)(x+3-2i)$. Complete the square: $(x+3)^2 - 9 + 13 = (x+3)^2 + 4$.
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What is $i^{10}$?
What is $i^{10}$?
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$i^{10}=-1$. $i^{10} = i^{8} \cdot i^2 = 1 \cdot (-1) = -1$
$i^{10}=-1$. $i^{10} = i^{8} \cdot i^2 = 1 \cdot (-1) = -1$
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What are the complex roots of $x^2+2x+5=0$?
What are the complex roots of $x^2+2x+5=0$?
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$x=-1+2i,-1-2i$. Use quadratic formula with discriminant $4 - 20 = -16$.
$x=-1+2i,-1-2i$. Use quadratic formula with discriminant $4 - 20 = -16$.
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Factor $x^2+2x+5$ over the complex numbers.
Factor $x^2+2x+5$ over the complex numbers.
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$(x+1+2i)(x+1-2i)$. Complete the square: $(x+1)^2 - 1 + 5 = (x+1)^2 + 4$.
$(x+1+2i)(x+1-2i)$. Complete the square: $(x+1)^2 - 1 + 5 = (x+1)^2 + 4$.
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What are the complex roots of $x^2-6x+13=0$?
What are the complex roots of $x^2-6x+13=0$?
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$x=3+2i,3-2i$. Use quadratic formula with discriminant $36 - 52 = -16$.
$x=3+2i,3-2i$. Use quadratic formula with discriminant $36 - 52 = -16$.
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Rewrite $x^2+25$ as a product of two complex conjugate binomials.
Rewrite $x^2+25$ as a product of two complex conjugate binomials.
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$(x+5i)(x-5i)$. $x^2 + 25 = x^2 + (5i)^2$ factors as conjugate pair.
$(x+5i)(x-5i)$. $x^2 + 25 = x^2 + (5i)^2$ factors as conjugate pair.
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Rewrite $x^2+49$ as a product of two complex conjugate binomials.
Rewrite $x^2+49$ as a product of two complex conjugate binomials.
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$(x+7i)(x-7i)$. $x^2 + 49 = x^2 + (7i)^2$ factors as conjugate pair.
$(x+7i)(x-7i)$. $x^2 + 49 = x^2 + (7i)^2$ factors as conjugate pair.
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Rewrite $x^2+64$ as a product of two complex conjugate binomials.
Rewrite $x^2+64$ as a product of two complex conjugate binomials.
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$(x+8i)(x-8i)$. $x^2 + 64 = x^2 + (8i)^2$ factors as conjugate pair.
$(x+8i)(x-8i)$. $x^2 + 64 = x^2 + (8i)^2$ factors as conjugate pair.
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Rewrite $x^2+100$ as a product of two complex conjugate binomials.
Rewrite $x^2+100$ as a product of two complex conjugate binomials.
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$(x+10i)(x-10i)$. $x^2 + 100 = x^2 + (10i)^2$ factors as conjugate pair.
$(x+10i)(x-10i)$. $x^2 + 100 = x^2 + (10i)^2$ factors as conjugate pair.
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Rewrite $x^2+2$ as a product of two complex conjugate binomials.
Rewrite $x^2+2$ as a product of two complex conjugate binomials.
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$(x+i\sqrt{2})(x-i\sqrt{2})$. $x^2 + 2 = x^2 + (i\sqrt{2})^2$ factors as conjugate pair.
$(x+i\sqrt{2})(x-i\sqrt{2})$. $x^2 + 2 = x^2 + (i\sqrt{2})^2$ factors as conjugate pair.
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Rewrite $x^2+7$ as a product of two complex conjugate binomials.
Rewrite $x^2+7$ as a product of two complex conjugate binomials.
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$(x+i\sqrt{7})(x-i\sqrt{7})$. $x^2 + 7 = x^2 + (i\sqrt{7})^2$ factors as conjugate pair.
$(x+i\sqrt{7})(x-i\sqrt{7})$. $x^2 + 7 = x^2 + (i\sqrt{7})^2$ factors as conjugate pair.
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Rewrite $x^2+12$ as a product of two complex conjugate binomials.
Rewrite $x^2+12$ as a product of two complex conjugate binomials.
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$(x+2i\sqrt{3})(x-2i\sqrt{3})$. $12 = 4 \cdot 3$, so $\sqrt{12} = 2\sqrt{3}$.
$(x+2i\sqrt{3})(x-2i\sqrt{3})$. $12 = 4 \cdot 3$, so $\sqrt{12} = 2\sqrt{3}$.
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Factor $x^2+4x+8$ over the complex numbers.
Factor $x^2+4x+8$ over the complex numbers.
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$(x+2+2i)(x+2-2i)$. Complete the square: $(x+2)^2 - 4 + 8 = (x+2)^2 + 4$.
$(x+2+2i)(x+2-2i)$. Complete the square: $(x+2)^2 - 4 + 8 = (x+2)^2 + 4$.
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Factor $x^2+10x+29$ over the complex numbers.
Factor $x^2+10x+29$ over the complex numbers.
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$(x+5+2i)(x+5-2i)$. Complete the square: $(x+5)^2 - 25 + 29 = (x+5)^2 + 4$.
$(x+5+2i)(x+5-2i)$. Complete the square: $(x+5)^2 - 25 + 29 = (x+5)^2 + 4$.
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What are the complex roots of $x^2+4=0$?
What are the complex roots of $x^2+4=0$?
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$x=2i,-2i$. Solve $x^2 = -4$ to get $x = \pm 2i$.
$x=2i,-2i$. Solve $x^2 = -4$ to get $x = \pm 2i$.
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What are the complex roots of $x^2-2x+2=0$?
What are the complex roots of $x^2-2x+2=0$?
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$x=1+i,1-i$. Use quadratic formula with discriminant $4 - 8 = -4$.
$x=1+i,1-i$. Use quadratic formula with discriminant $4 - 8 = -4$.
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State the Complex Conjugate Root Theorem for polynomials with real coefficients.
State the Complex Conjugate Root Theorem for polynomials with real coefficients.
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If $a+bi$ is a root, then $a-bi$ is a root. Complex roots of real polynomials come in conjugate pairs.
If $a+bi$ is a root, then $a-bi$ is a root. Complex roots of real polynomials come in conjugate pairs.
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What is the product $(a+bi)(a-bi)$ equal to, simplified?
What is the product $(a+bi)(a-bi)$ equal to, simplified?
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$a^2+b^2$. Multiplying complex conjugates eliminates the imaginary terms.
$a^2+b^2$. Multiplying complex conjugates eliminates the imaginary terms.
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Identify the factorization of $x^2-2ax+(a^2+b^2)$ over complex numbers.
Identify the factorization of $x^2-2ax+(a^2+b^2)$ over complex numbers.
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$(x-(a+bi))(x-(a-bi))$. Standard form with roots $a \pm bi$.
$(x-(a+bi))(x-(a-bi))$. Standard form with roots $a \pm bi$.
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What is the factored form of $x^2+2ax+(a^2+b^2)$ over complex numbers?
What is the factored form of $x^2+2ax+(a^2+b^2)$ over complex numbers?
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$(x-(-a+bi))(x-(-a-bi))$. Standard form with roots $-a \pm bi$.
$(x-(-a+bi))(x-(-a-bi))$. Standard form with roots $-a \pm bi$.
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Rewrite $x^2+3$ as a product of two complex conjugate binomials.
Rewrite $x^2+3$ as a product of two complex conjugate binomials.
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$(x+i\sqrt{3})(x-i\sqrt{3})$. $x^2 + 3 = x^2 + (i\sqrt{3})^2$ factors as conjugate pair.
$(x+i\sqrt{3})(x-i\sqrt{3})$. $x^2 + 3 = x^2 + (i\sqrt{3})^2$ factors as conjugate pair.
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State the identity for factoring a sum of squares using $i$.
State the identity for factoring a sum of squares using $i$.
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$a^2+b^2=(a+bi)(a-bi)$. Uses complex conjugates to factor sum of squares.
$a^2+b^2=(a+bi)(a-bi)$. Uses complex conjugates to factor sum of squares.
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What is $i^3$?
What is $i^3$?
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$i^3=-i$. $i^3 = i^2 \cdot i = -1 \cdot i = -i$
$i^3=-i$. $i^3 = i^2 \cdot i = -1 \cdot i = -i$
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Rewrite $x^2+4$ as a product of two complex conjugate binomials.
Rewrite $x^2+4$ as a product of two complex conjugate binomials.
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$(x+2i)(x-2i)$. $x^2 + 4 = x^2 + (2i)^2$ factors as conjugate pair.
$(x+2i)(x-2i)$. $x^2 + 4 = x^2 + (2i)^2$ factors as conjugate pair.
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Rewrite $x^2+9$ as a product of two complex conjugate binomials.
Rewrite $x^2+9$ as a product of two complex conjugate binomials.
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$(x+3i)(x-3i)$. $x^2 + 9 = x^2 + (3i)^2$ factors as conjugate pair.
$(x+3i)(x-3i)$. $x^2 + 9 = x^2 + (3i)^2$ factors as conjugate pair.
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Rewrite $x^2+1$ as a product of two complex conjugate binomials.
Rewrite $x^2+1$ as a product of two complex conjugate binomials.
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$(x+i)(x-i)$. $x^2 + 1 = x^2 + i^2$ factors as conjugate pair.
$(x+i)(x-i)$. $x^2 + 1 = x^2 + i^2$ factors as conjugate pair.
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Rewrite $x^2+16$ as a product of two complex conjugate binomials.
Rewrite $x^2+16$ as a product of two complex conjugate binomials.
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$(x+4i)(x-4i)$. $x^2 + 16 = x^2 + (4i)^2$ factors as conjugate pair.
$(x+4i)(x-4i)$. $x^2 + 16 = x^2 + (4i)^2$ factors as conjugate pair.
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Factor $x^2-4x+5$ over the complex numbers.
Factor $x^2-4x+5$ over the complex numbers.
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$(x-2+i)(x-2-i)$. Complete the square: $(x-2)^2 - 4 + 5 = (x-2)^2 + 1$.
$(x-2+i)(x-2-i)$. Complete the square: $(x-2)^2 - 4 + 5 = (x-2)^2 + 1$.
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