Algebra 2 Flashcards: Extending Polynomial Identities To Complex Numbers

What this deck covers

This deck focuses on Extending Polynomial Identities To Complex Numbers, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Rewrite $x^2+36$ as a product of two complex conjugate binomials.

Answer: $(x+6i)(x-6i)$. $x^2 + 36 = x^2 + (6i)^2$ factors as conjugate pair.

Flashcard 2: Rewrite $x^2+5$ as a product of two complex conjugate binomials.

Answer: $(x+i\sqrt{5})(x-i\sqrt{5})$. $x^2 + 5 = x^2 + (i\sqrt{5})^2$ factors as conjugate pair.

Flashcard 3: What is $i^2$?

Answer: $i^2=-1$. Definition of the imaginary unit.

Flashcard 4: What is $i^{27}$?

Answer: $i^{27}=-i$. $i^{27} = i^{24} \cdot i^3 = 1 \cdot (-i) = -i$

Flashcard 5: Rewrite $x^2+50$ as a product of two complex conjugate binomials.

Answer: $(x+5i\sqrt{2})(x-5i\sqrt{2})$. $50 = 25 \cdot 2$, so $\sqrt{50} = 5\sqrt{2}$.

Flashcard 6: What is $i^4$?

Answer: $i^4=1$. $i^4 = (i^2)^2 = (-1)^2 = 1$

Flashcard 7: What are the complex roots of $x^2-2x+2=0$?

Answer: $x=1+i,1-i$. Use quadratic formula with discriminant $4 - 8 = -4$.

Flashcard 8: Factor $x^2+2x+5$ over the complex numbers.

Answer: $(x+1+2i)(x+1-2i)$. Complete the square: $(x+1)^2 - 1 + 5 = (x+1)^2 + 4$.

Flashcard 9: What are the complex roots of $x^2+9=0$?

Answer: $x=3i,-3i$. Solve $x^2 = -9$ to get $x = \pm 3i$.

Flashcard 10: Rewrite $x^2+2$ as a product of two complex conjugate binomials.

Answer: $(x+i\sqrt{2})(x-i\sqrt{2})$. $x^2 + 2 = x^2 + (i\sqrt{2})^2$ factors as conjugate pair.

Flashcard 11: What is $i^{100}$?

Answer: $i^{100}=1$. $i^{100} = (i^4)^{25} = 1^{25} = 1$

Flashcard 12: Factor $x^2+4x+8$ over the complex numbers.

Answer: $(x+2+2i)(x+2-2i)$. Complete the square: $(x+2)^2 - 4 + 8 = (x+2)^2 + 4$.

Flashcard 13: Factor $x^2+8x+20$ over the complex numbers.

Answer: $(x+4+2i)(x+4-2i)$. Complete the square: $(x+4)^2 - 16 + 20 = (x+4)^2 + 4$.

Flashcard 14: Identify the factorization of $x^2-2ax+(a^2+b^2)$ over complex numbers.

Answer: $(x-(a+bi))(x-(a-bi))$. Standard form with roots $a \pm bi$.

Flashcard 15: What is $i^3$?

Answer: $i^3=-i$. $i^3 = i^2 \cdot i = -1 \cdot i = -i$

Flashcard 16: Rewrite $x^2+27$ as a product of two complex conjugate binomials.

Answer: $(x+3i\sqrt{3})(x-3i\sqrt{3})$. $27 = 9 \cdot 3$, so $\sqrt{27} = 3\sqrt{3}$.

Flashcard 17: Rewrite $x^2+7$ as a product of two complex conjugate binomials.

Answer: $(x+i\sqrt{7})(x-i\sqrt{7})$. $x^2 + 7 = x^2 + (i\sqrt{7})^2$ factors as conjugate pair.

Flashcard 18: State the identity for factoring a difference of squares over complex numbers.

Answer: $a^2-b^2=(a-b)(a+b)$. Standard difference of squares identity, valid for complex numbers.

Flashcard 19: What are the complex roots of $x^2+1=0$?

Answer: $x=i,-i$. Solve $x^2 = -1$ to get $x = \pm i$.

Flashcard 20: Factor $x^2-6x+13$ over the complex numbers.

Answer: $(x-3+2i)(x-3-2i)$. Complete the square: $(x-3)^2 - 9 + 13 = (x-3)^2 + 4$.

Flashcard 21: What are the complex roots of $x^2-6x+13=0$?

Answer: $x=3+2i,3-2i$. Use quadratic formula with discriminant $36 - 52 = -16$.

Flashcard 22: Rewrite $x^2+64$ as a product of two complex conjugate binomials.

Answer: $(x+8i)(x-8i)$. $x^2 + 64 = x^2 + (8i)^2$ factors as conjugate pair.

Flashcard 23: Rewrite $x^2+12$ as a product of two complex conjugate binomials.

Answer: $(x+2i\sqrt{3})(x-2i\sqrt{3})$. $12 = 4 \cdot 3$, so $\sqrt{12} = 2\sqrt{3}$.

Flashcard 24: Rewrite $x^2+25$ as a product of two complex conjugate binomials.

Answer: $(x+5i)(x-5i)$. $x^2 + 25 = x^2 + (5i)^2$ factors as conjugate pair.

Flashcard 25: What are the complex roots of $x^2+4=0$?

Answer: $x=2i,-2i$. Solve $x^2 = -4$ to get $x = \pm 2i$.

Flashcard 26: Factor $x^2+10x+29$ over the complex numbers.

Answer: $(x+5+2i)(x+5-2i)$. Complete the square: $(x+5)^2 - 25 + 29 = (x+5)^2 + 4$.

Flashcard 27: Factor $x^2-2x+2$ over the complex numbers.

Answer: $(x-1+i)(x-1-i)$. Complete the square: $(x-1)^2 - 1 + 2 = (x-1)^2 + 1$.

Flashcard 28: Factor $x^2-10x+29$ over the complex numbers.

Answer: $(x-5+2i)(x-5-2i)$. Complete the square: $(x-5)^2 - 25 + 29 = (x-5)^2 + 4$.

Flashcard 29: What are the complex roots of $x^2+2x+5=0$?

Answer: $x=-1+2i,-1-2i$. Use quadratic formula with discriminant $4 - 20 = -16$.

Flashcard 30: Rewrite $x^2+1$ as a product of two complex conjugate binomials.

Answer: $(x+i)(x-i)$. $x^2 + 1 = x^2 + i^2$ factors as conjugate pair.

Flashcard 31: What is the factored form of $x^2+2ax+(a^2+b^2)$ over complex numbers?

Answer: $(x-(-a+bi))(x-(-a-bi))$. Standard form with roots $-a \pm bi$.

Flashcard 32: Factor $x^2+6x+13$ over the complex numbers.

Answer: $(x+3+2i)(x+3-2i)$. Complete the square: $(x+3)^2 - 9 + 13 = (x+3)^2 + 4$.

Flashcard 33: Rewrite $x^2+49$ as a product of two complex conjugate binomials.

Answer: $(x+7i)(x-7i)$. $x^2 + 49 = x^2 + (7i)^2$ factors as conjugate pair.

Flashcard 34: Rewrite $x^2+8$ as a product of two complex conjugate binomials.

Answer: $(x+2i\sqrt{2})(x-2i\sqrt{2})$. $8 = 4 \cdot 2$, so $\sqrt{8} = 2\sqrt{2}$.

Flashcard 35: Rewrite $x^2+4$ as a product of two complex conjugate binomials.

Answer: $(x+2i)(x-2i)$. $x^2 + 4 = x^2 + (2i)^2$ factors as conjugate pair.

Flashcard 36: State the Complex Conjugate Root Theorem for polynomials with real coefficients.

Answer: If $a+bi$ is a root, then $a-bi$ is a root. Complex roots of real polynomials come in conjugate pairs.

Flashcard 37: Factor $x^2+4x+5$ over the complex numbers.

Answer: $(x+2+i)(x+2-i)$. Complete the square: $(x+2)^2 - 4 + 5 = (x+2)^2 + 1$.

Flashcard 38: What are the complex roots of $x^2+16=0$?

Answer: $x=4i,-4i$. Solve $x^2 = -16$ to get $x = \pm 4i$.

Flashcard 39: Rewrite $x^2+3$ as a product of two complex conjugate binomials.

Answer: $(x+i\sqrt{3})(x-i\sqrt{3})$. $x^2 + 3 = x^2 + (i\sqrt{3})^2$ factors as conjugate pair.

Flashcard 40: Factor $x^2-4x+8$ over the complex numbers.

Answer: $(x-2+2i)(x-2-2i)$. Complete the square: $(x-2)^2 - 4 + 8 = (x-2)^2 + 4$.

Flashcard 41: What are the complex roots of $x^2+2=0$?

Answer: $x=i\sqrt{2},-i\sqrt{2}$. Solve $x^2 = -2$ to get $x = \pm i\sqrt{2}$.

Flashcard 42: State the identity for factoring a sum of squares using $i$.

Answer: $a^2+b^2=(a+bi)(a-bi)$. Uses complex conjugates to factor sum of squares.

Flashcard 43: Factor $x^2-4x+5$ over the complex numbers.

Answer: $(x-2+i)(x-2-i)$. Complete the square: $(x-2)^2 - 4 + 5 = (x-2)^2 + 1$.

Flashcard 44: What is $i^{10}$?

Answer: $i^{10}=-1$. $i^{10} = i^{8} \cdot i^2 = 1 \cdot (-1) = -1$

Flashcard 45: What is the product $(a+bi)(a-bi)$ equal to, simplified?

Answer: $a^2+b^2$. Multiplying complex conjugates eliminates the imaginary terms.

Flashcard 46: Rewrite $x^2+18$ as a product of two complex conjugate binomials.

Answer: $(x+3i\sqrt{2})(x-3i\sqrt{2})$. $18 = 9 \cdot 2$, so $\sqrt{18} = 3\sqrt{2}$.

Flashcard 47: Rewrite $x^2+9$ as a product of two complex conjugate binomials.

Answer: $(x+3i)(x-3i)$. $x^2 + 9 = x^2 + (3i)^2$ factors as conjugate pair.

Flashcard 48: Rewrite $x^2+16$ as a product of two complex conjugate binomials.

Answer: $(x+4i)(x-4i)$. $x^2 + 16 = x^2 + (4i)^2$ factors as conjugate pair.

Flashcard 49: Rewrite $x^2+100$ as a product of two complex conjugate binomials.

Answer: $(x+10i)(x-10i)$. $x^2 + 100 = x^2 + (10i)^2$ factors as conjugate pair.

Flashcard 50: Factor $x^2-8x+20$ over the complex numbers.

Answer: $(x-4+2i)(x-4-2i)$. Complete the square: $(x-4)^2 - 16 + 20 = (x-4)^2 + 4$.