Algebra 2 Flashcards: Fundamental Theorem Of Algebra For Quadratics

What this deck covers

This deck focuses on Fundamental Theorem Of Algebra For Quadratics, 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: Given roots $r_1=5$ and $r_2=-1$, what is the monic quadratic polynomial?

Answer: $x^2-4x-5$. Using sum=$4$ and product=$-5$ in $x^2-(sum)x+(product)$.

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

Answer: $x=i$ and $x=-i$. Solving $x^2=-1$ gives $x=\pm\sqrt{-1}=\pm i$.

Flashcard 3: For $x^2+px+q=0$, what are the sum and product of the roots?

Answer: $r_1+r_2=-p$ and $r_1r_2=q$. For monic quadratic $x^2+px+q$, coefficients relate directly to roots.

Flashcard 4: Write the monic quadratic with roots $-4+i$ and $-4-i$.

Answer: $x^2+8x+17$. Using Vieta's formulas: sum=$-8$, product=$17$.

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

Answer: $(x-(1+2i))(x-(1-2i))$. Using roots $1+2i$ and $1-2i$ to form linear factors.

Flashcard 6: What is the relationship between complex roots and linear factors over $\mathbb{C}$ for a quadratic?

Answer: A quadratic factors into two linear factors over $\mathbb{C}$. FTA guarantees complete factorization over complex numbers.

Flashcard 7: If $\Delta<0$ in $x=\frac{-b\pm\sqrt{\Delta}}{2a}$, what form do the two solutions take?

Answer: $x=\frac{-b}{2a}\pm i\frac{\sqrt{-\Delta}}{2a}$. Standard form showing complex conjugate pair structure.

Flashcard 8: If $ax^2+bx+c$ has roots $r_1,r_2$, what is $c$ in terms of $a$ and $r_1r_2$?

Answer: $c=a(r_1r_2)$. From Vieta's formulas, product of roots equals $\frac{c}{a}$.

Flashcard 9: Write $x^2+6x+9$ in factored form showing multiplicity.

Answer: $(x+3)^2$. Perfect square trinomial factors as $(x+3)^2$.

Flashcard 10: Which discriminant condition gives two distinct real roots for $ax^2+bx+c=0$?

Answer: $\Delta>0$. Positive discriminant means the square root yields two real values.

Flashcard 11: What is the degree of a quadratic polynomial written as $ax^2+bx+c$ with $a\ne 0$?

Answer: Degree $2$. The highest power term $ax^2$ determines the degree is $2$.

Flashcard 12: Which discriminant condition gives exactly one real root (a repeated root) for $ax^2+bx+c=0$?

Answer: $\Delta=0$. Zero discriminant makes the $\pm$ term disappear, giving one solution.

Flashcard 13: Find the complex roots of $x^2-2x+2=0$.

Answer: $x=1\pm i$. Using quadratic formula with $\Delta=-4<0$.

Flashcard 14: Which discriminant condition gives two nonreal complex roots for $ax^2+bx+c=0$?

Answer: $\Delta<0$. Negative discriminant requires imaginary unit for square root.

Flashcard 15: What is the repeated root of $x^2+6x+9=0$?

Answer: $x=-3$. Since $\Delta=0$, there's one repeated root.

Flashcard 16: What does it mean for a root of a polynomial to have multiplicity $2$?

Answer: The factor $(x-r)^2$ divides the polynomial. Multiplicity $2$ means the root appears twice in factorization.

Flashcard 17: What is the standard factorization form of a quadratic with roots $r_1$ and $r_2$?

Answer: $a(x-r_1)(x-r_2)$. Each root contributes a linear factor $(x-r_i)$.

Flashcard 18: Find the complex roots of $x^2+10x+34=0$.

Answer: $x=-5\pm 3i$. Using quadratic formula with $\Delta=-36<0$.

Flashcard 19: Find the discriminant $\Delta$ of $x^2+6x+9=0$.

Answer: $\Delta=0$. Using $\Delta=b^2-4ac$ with $a=1,b=6,c=9$.

Flashcard 20: Factor $x^2+2x+5$ over the complex numbers.

Answer: $(x-(-1+2i))(x-(-1-2i))$. Using roots $-1+2i$ and $-1-2i$ to form linear factors.

Flashcard 21: Find the roots of $7x^2-1=0$.

Answer: $x=\pm\frac{1}{\sqrt{7}}$. Solving $7x^2=1$ gives $x^2=\frac{1}{7}$.

Flashcard 22: How many real roots does $x^2-4x+7=0$ have, based on $\Delta$?

Answer: $0$ real roots. Since $\Delta<0$, there are no real roots.

Flashcard 23: If $ax^2+bx+c$ has roots $r_1,r_2$, what is $b$ in terms of $a$ and $r_1+r_2$?

Answer: $b=-a(r_1+r_2)$. From Vieta's formulas, sum of roots equals $-\frac{b}{a}$.

Flashcard 24: Find $r_1+r_2$ for $3x^2-12x+1=0$ without solving.

Answer: $r_1+r_2=4$. Sum formula: $r_1+r_2=-\frac{b}{a}=-\frac{-12}{3}=4$.

Flashcard 25: What is the factored form of $x^2-2ax+(a^2+b^2)$ over complex numbers?

Answer: $(x-(a+bi))(x-(a-bi))$. Linear factors corresponding to conjugate pair roots.

Flashcard 26: What does the Complex Conjugate Root Theorem say for polynomials with real coefficients?

Answer: If $a+bi$ is a root, then $a-bi$ is a root. Real coefficients force complex roots to come in conjugate pairs.

Flashcard 27: Find the roots of $4x^2+4x+5=0$ in $a\pm bi$ form.

Answer: $x=-\frac{1}{2}\pm i$. Using quadratic formula with $\Delta=-64<0$.

Flashcard 28: Find the complex solutions of $x^2-4x+7=0$ in $a\pm bi$ form.

Answer: $x=2\pm i\sqrt{3}$. Using quadratic formula with $\Delta=-12<0$.

Flashcard 29: Identify the complex roots of $x^2+9=0$.

Answer: $x=3i$ and $x=-3i$. Solving $x^2=-9$ gives $x=\pm\sqrt{-9}=\pm 3i$.

Flashcard 30: Find the solutions of $2x^2+3x-2=0$.

Answer: $x=\frac{1}{2}$ and $x=-2$. Using quadratic formula with $\Delta=25>0$.

Flashcard 31: What does it mean to count roots "with multiplicity" for a quadratic?

Answer: A double root counts as $2$ roots total. Repeated roots are counted multiple times in the total.

Flashcard 32: Find $r_1r_2$ for $3x^2-12x+1=0$ without solving.

Answer: $r_1r_2=\frac{1}{3}$. Product formula: $r_1r_2=\frac{c}{a}=\frac{1}{3}$.

Flashcard 33: What is the quadratic formula for the solutions of $ax^2+bx+c=0$?

Answer: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Standard formula derived from completing the square method.

Flashcard 34: What does the Fundamental Theorem of Algebra say about the number of complex roots of a degree $n$ polynomial (counting multiplicity)?

Answer: Exactly $n$ complex roots, counting multiplicity. FTA states degree $n$ polynomial has exactly $n$ complex roots when counting multiplicity.

Flashcard 35: Identify the missing statement to show FTA for quadratics: $\Delta<0\Rightarrow\sqrt{\Delta}=i\sqrt{-\Delta}$.

Answer: $\sqrt{\Delta}=i\sqrt{-\Delta}$. When discriminant is negative, square root becomes imaginary.

Flashcard 36: Find the complex roots of $x^2-8x+20=0$.

Answer: $x=4\pm 2i$. Using quadratic formula with $\Delta=-16<0$.

Flashcard 37: What does the Fundamental Theorem of Algebra guarantee for any nonconstant polynomial with complex coefficients?

Answer: It has at least one complex root. FTA guarantees at least one complex root for any nonconstant polynomial.

Flashcard 38: What is the definition of the imaginary unit $i$?

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

Flashcard 39: How do you rewrite $\sqrt{-k}$ for $k>0$ using $i$?

Answer: $\sqrt{-k}=i\sqrt{k}$. Standard method to express square roots of negative numbers.

Flashcard 40: Identify the conjugate of the complex root $3-5i$.

Answer: $3+5i$. Complex conjugate of $3-5i$ flips the imaginary sign.

Flashcard 41: What is the conjugate of a complex number $a+bi$?

Answer: $a-bi$. Complex conjugate changes the sign of the imaginary part.

Flashcard 42: What are the sum and product of roots for $ax^2+bx+c=0$ in terms of $a,b,c$?

Answer: $r_1+r_2=-\frac{b}{a}$ and $r_1r_2=\frac{c}{a}$. Vieta's formulas relate coefficients to root combinations.

Flashcard 43: If a real-coefficient quadratic has root $-1+2i$, what is the other root?

Answer: $-1-2i$. Complex Conjugate Root Theorem for real coefficients.

Flashcard 44: Find the discriminant $\Delta$ of $2x^2+3x-2=0$.

Answer: $\Delta=25$. Using $\Delta=b^2-4ac$ with $a=2,b=3,c=-2$.

Flashcard 45: What does the Fundamental Theorem of Algebra imply about the total number of complex zeros of a quadratic (counting multiplicity)?

Answer: It has $2$ complex zeros, counting multiplicity. Since degree is $2$, FTA guarantees exactly $2$ complex zeros.

Flashcard 46: Identify the number of complex roots (counting multiplicity) of $p(x)=7x^2-1$.

Answer: $2$. Degree $2$ polynomial always has exactly $2$ complex roots.

Flashcard 47: What does the Factor Theorem state about $r$ and the factor $(x-r)$?

Answer: $p(r)=0\iff(x-r)$ is a factor of $p(x)$. Fundamental connection between roots and linear factors.

Flashcard 48: What is the discriminant of $ax^2+bx+c$?

Answer: $\Delta=b^2-4ac$. The discriminant determines the nature of the roots.

Flashcard 49: For $x^2-2ax+(a^2+b^2)=0$, what are the roots in terms of $a$ and $b$?

Answer: $x=a\pm bi$. Standard form for conjugate pair roots $a\pm bi$.

Flashcard 50: Find the roots of $x^2-6x+13=0$ in $a\pm bi$ form.

Answer: $x=3\pm 2i$. Using quadratic formula with discriminant $\Delta=-16<0$.

Flashcard 51: Find the discriminant $\Delta$ of $x^2-4x+7=0$.

Answer: $\Delta=-12$. Using $\Delta=b^2-4ac$ with $a=1,b=-4,c=7$.

Flashcard 52: Write the monic quadratic with roots $2+3i$ and $2-3i$.

Answer: $x^2-4x+13$. Using Vieta's formulas: sum=$4$, product=$13$.

Flashcard 53: Find the complex roots of $x^2+2x+2=0$.

Answer: $x=-1\pm i$. Using quadratic formula with $\Delta=-4<0$.

Flashcard 54: Find the complex solutions of $x^2+4x+8=0$ in $a\pm bi$ form.

Answer: $x=-2\pm 2i$. Using quadratic formula with negative discriminant.

Flashcard 55: What is the key conclusion that verifies the Fundamental Theorem of Algebra for quadratics?

Answer: Every quadratic has $2$ complex roots, counting multiplicity. This confirms FTA applies to all degree $2$ polynomials.

Flashcard 56: What does it mean for a complex number to be a root of $p(x)$?

Answer: $p(r)=0$. A root makes the polynomial evaluate to zero.