AP Precalculus Flashcards: Polynomial Functions And Complex Zeros

What this deck covers

This deck focuses on Polynomial Functions And Complex Zeros, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

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: What is the modulus of $3 + 4i$?

Answer:

1. Use $|a + bi| = \sqrt{a^2 + b^2} = \sqrt{9 + 16}$.

Flashcard 2: What is the complex conjugate of $7 - 5i$?

Answer: $7 + 5i$. Change the sign of the imaginary part.

Flashcard 3: Find the complex roots of $x^2 + 4$.

Answer: $x = 2i, -2i$. Set $x^2 = -4$ and solve for $x = \pm 2i$.

Flashcard 4: Find the zeros of $x^2 - 9$.

Answer: $x = 3, -3$. Factor as $(x-3)(x+3)$ and set each factor to zero.

Flashcard 5: Find the zeros of $x^2 - 9$.

Answer: $x = 3, -3$. Factor as $(x-3)(x+3)$ and set each factor to zero.

Flashcard 6: What is the imaginary part of $5 + 6i$?

Answer:

1. The coefficient of $i$ in the complex number.

Flashcard 7: Determine the degree of $3x^4 + 2x^2 - 7$.

Answer:

1. The highest power term determines the degree.

Flashcard 8: What is the factor theorem?

Answer: $x - c$ is a factor if $f(c) = 0$. Direct consequence of the remainder theorem when remainder is zero.

Flashcard 9: State the definition of a root of multiplicity.

Answer: A root repeated in the factorization of a polynomial. Indicates how many times a root appears in the factorization.

Flashcard 10: Simplify $(1 + 2i)^2$.

Answer: $-3 + 4i$. Expand: $(1+2i)^2 = 1 + 4i + 4i^2 = 1 + 4i - 4$.

Flashcard 11: Simplify $(1 + 2i)^2$.

Answer: $-3 + 4i$. Expand: $(1+2i)^2 = 1 + 4i + 4i^2 = 1 + 4i - 4$.

Flashcard 12: Find the zero of $x^2 - 4x + 4$.

Answer: $x = 2$. This is $(x-2)^2$, so $x = 2$ is a repeated root.

Flashcard 13: What is Descartes' Rule of Signs?

Answer: Predicts the number of positive and negative real roots. Counts sign changes to estimate the number of positive/negative roots.

Flashcard 14: State the Remainder Theorem.

Answer: The remainder of $f(x)$ divided by $x - c$ is $f(c)$. Fundamental theorem for polynomial division and evaluation.

Flashcard 15: What is a polynomial's leading coefficient?

Answer: Coefficient of the highest degree term. Same as leading coefficient - determines polynomial behavior.

Flashcard 16: Find the product $(3 + 2i)(3 - 2i)$.

Answer:

1. Conjugate pairs multiply to give $a^2 + b^2 = 9 + 4 = 13$.

Flashcard 17: State the Remainder Theorem.

Answer: The remainder of $f(x)$ divided by $x - c$ is $f(c)$. Fundamental theorem for polynomial division and evaluation.

Flashcard 18: Find the product $(3 + 2i)(3 - 2i)$.

Answer:

1. Conjugate pairs multiply to give $a^2 + b^2 = 9 + 4 = 13$.

Flashcard 19: State the Conjugate Root Theorem.

Answer: If $a + bi$ is a root, $a - bi$ is also a root. For polynomials with real coefficients, complex roots come in conjugate pairs.

Flashcard 20: What is the degree of a polynomial?

Answer: The highest power of the variable in the polynomial. Determines the polynomial's behavior and number of roots.

Flashcard 21: State the definition of a monomial.

Answer: A polynomial with one term. Simplest polynomial with exactly one term.

Flashcard 22: What is the multiplicity of zero for $(x - 2)^3$?

Answer: $3$. The exponent indicates how many times the zero appears.

Flashcard 23: Find the zero of $x^2 - 4x + 4$.

Answer: $x = 2$. This is $(x-2)^2$, so $x = 2$ is a repeated root.

Flashcard 24: What is the zero of a polynomial function?

Answer: A value $x$ for which $f(x) = 0$. Values where the polynomial equals zero.

Flashcard 25: State the Fundamental Theorem of Algebra.

Answer: Every non-zero polynomial has at least one complex root. This guarantees that all polynomial equations have solutions in the complex number system.

Flashcard 26: Find the sum of the roots of $x^2 - 5x + 6$.

Answer:

1. For $ax^2 + bx + c$, sum of roots equals $-b/a = 5/1$.

Flashcard 27: What is the degree of a polynomial?

Answer: The highest power of the variable in the polynomial. Determines the polynomial's behavior and number of roots.

Flashcard 28: What is the degree of the polynomial $0$?

Answer: Undefined. The zero polynomial has no degree by convention.

Flashcard 29: What is the degree of the zero polynomial?

Answer: Undefined. The zero polynomial has no terms, so degree is undefined.

Flashcard 30: Calculate the modulus of $1 - i$.

Answer: $\text{sqrt}(2)$. Use $|1-i| = \sqrt{1^2 + (-1)^2} = \sqrt{2}$.

Flashcard 31: What is the multiplicity of zero for $(x - 2)^3$?

Answer:

1. The exponent indicates how many times the zero appears.

Flashcard 32: What is the complex conjugate of $7 - 5i$?

Answer: $7 + 5i$. Change the sign of the imaginary part.

Flashcard 33: What is the real part of $4 - 3i$?

Answer:

1. The coefficient of the term without $i$.

Flashcard 34: Identify the conjugate of $3 + 4i$.

Answer: $3 - 4i$. Change the sign of the imaginary part to get the conjugate.

Flashcard 35: Calculate the discriminant of $x^2 + 4x + 5$.

Answer: $-4$. Use $b^2 - 4ac = 16 - 20 = -4$ for negative discriminant.

Flashcard 36: State the Conjugate Root Theorem.

Answer: If $a + bi$ is a root, $a - bi$ is also a root. For polynomials with real coefficients, complex roots come in conjugate pairs.

Flashcard 37: Define a complex number.

Answer: A number in the form $a + bi$, where $a$ and $b$ are real numbers. Standard form where $i = \sqrt{-1}$ represents the imaginary unit.

Flashcard 38: State the definition of a monomial.

Answer: A polynomial with one term. Simplest polynomial with exactly one term.

Flashcard 39: What does the term 'roots' refer to in polynomials?

Answer: Values that satisfy $f(x) = 0$. Same as zeros - values that make the polynomial equal zero.

Flashcard 40: What is the real part of $4 - 3i$?

Answer:

1. The coefficient of the term without $i$.

Flashcard 41: What is the modulus of $3 + 4i$?

Answer:

1. Use $|a + bi| = \sqrt{a^2 + b^2} = \sqrt{9 + 16}$.

Flashcard 42: Calculate the modulus of $1 - i$.

Answer: $\text{sqrt}(2)$. Use $|1-i| = \sqrt{1^2 + (-1)^2} = \sqrt{2}$.

Flashcard 43: What is the zero of a polynomial function?

Answer: A value $x$ for which $f(x) = 0$. Values where the polynomial equals zero.

Flashcard 44: What is a polynomial's leading coefficient?

Answer: Coefficient of the highest degree term. Same as leading coefficient - determines polynomial behavior.

Flashcard 45: What is the imaginary unit $i$?

Answer: $i = \text{sqrt}(-1)$. Fundamental unit for complex numbers, where $i^2 = -1$.

Flashcard 46: Identify the zeros of $x^2 + 1$.

Answer: $x = i, -i$. Set $x^2 = -1$ and solve for $x = \pm i$.

Flashcard 47: What is Descartes' Rule of Signs?

Answer: Predicts the number of positive and negative real roots. Counts sign changes to estimate the number of positive/negative roots.

Flashcard 48: Find the product of $2 + i$ and $2 - i$.

Answer:

1. Conjugate pairs multiply to $a^2 + b^2 = 4 + 1 = 5$.

Flashcard 49: Find the product of $2 + i$ and $2 - i$.

Answer:

1. Conjugate pairs multiply to $a^2 + b^2 = 4 + 1 = 5$.

Flashcard 50: What is the factor theorem?

Answer: $x - c$ is a factor if $f(c) = 0$. Direct consequence of the remainder theorem when remainder is zero.

Flashcard 51: Calculate the discriminant of $x^2 + 4x + 5$.

Answer: $-4$. Use $b^2 - 4ac = 16 - 20 = -4$ for negative discriminant.

Flashcard 52: What is the degree of the polynomial $0$?

Answer: Undefined. The zero polynomial has no degree by convention.

Flashcard 53: Find the complex roots of $x^2 + 4$.

Answer: $x = 2i, -2i$. Set $x^2 = -4$ and solve for $x = \pm 2i$.

Flashcard 54: Identify the zeros of $x^2 + 1$.

Answer: $x = i, -i$. Set $x^2 = -1$ and solve for $x = \pm i$.

Flashcard 55: Define a complex number.

Answer: A number in the form $a + bi$, where $a$ and $b$ are real numbers. Standard form where $i = \sqrt{-1}$ represents the imaginary unit.

Flashcard 56: Find the roots of $x^2 - 2x + 1$.

Answer: $x = 1$. This is $(x-1)^2$, so $x = 1$ is a repeated root.

Flashcard 57: State the Fundamental Theorem of Algebra.

Answer: Every non-zero polynomial has at least one complex root. This guarantees that all polynomial equations have solutions in the complex number system.

Flashcard 58: What is the standard form of a polynomial?

Answer: Terms arranged in descending order of exponents. Conventional ordering from highest to lowest degree terms.

Flashcard 59: State the definition of a root of multiplicity.

Answer: A root repeated in the factorization of a polynomial. Indicates how many times a root appears in the factorization.

Flashcard 60: What is the degree of $7x^3 + 2x^2 + x + 5$?

Answer:

1. The highest power of $x$ determines the degree.

Flashcard 61: What does the term 'roots' refer to in polynomials?

Answer: Values that satisfy $f(x) = 0$. Same as zeros - values that make the polynomial equal zero.

Flashcard 62: State the meaning of 'complex conjugate'.

Answer: Change the sign of the imaginary part of a complex number. Flip the sign of the imaginary part only.

Flashcard 63: Determine the degree of $3x^4 + 2x^2 - 7$.

Answer:

1. The highest power term determines the degree.

Flashcard 64: What is the leading term of $2x^5 + 3x^4$?

Answer: $2x^5$. The term with the highest degree in the polynomial.

Flashcard 65: What is a leading coefficient?

Answer: The coefficient of the term with the highest degree. Determines the polynomial's end behavior and properties.

Flashcard 66: What is the degree of the zero polynomial?

Answer: Undefined. The zero polynomial has no terms, so degree is undefined.

Flashcard 67: What is the imaginary part of $5 + 6i$?

Answer:

1. The coefficient of $i$ in the complex number.

Flashcard 68: What is the leading term of $2x^5 + 3x^4$?

Answer: $2x^5$. The term with the highest degree in the polynomial.