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 complex conjugate of $7 - 5i$?

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

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

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

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

Answer:

1. The highest power term determines the degree.

Flashcard 4: 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 5: Simplify $(1 + 2i)^2$.

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

Flashcard 6: Find the zero of $x^2 - 4x + 4$.

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

Flashcard 7: 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 8: Find the product $(3 + 2i)(3 - 2i)$.

Answer:

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

Flashcard 9: 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 10: State the definition of a monomial.

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

Flashcard 11: What is the multiplicity of zero for $(x - 2)^3$?

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

Flashcard 12: What is the zero of a polynomial function?

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

Flashcard 13: 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 14: 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 15: 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 16: What is the degree of the zero polynomial?

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

Flashcard 17: Calculate the modulus of $1 - i$.

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

Flashcard 18: What is the multiplicity of zero for $(x - 2)^3$?

Answer:

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

Flashcard 19: What is the real part of $4 - 3i$?

Answer:

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

Flashcard 20: Identify the conjugate of $3 + 4i$.

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

Flashcard 21: 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 22: 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 23: What is the modulus of $3 + 4i$?

Answer:

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

Flashcard 24: What is a polynomial's leading coefficient?

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

Flashcard 25: What is the imaginary unit $i$?

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

Flashcard 26: Identify the zeros of $x^2 + 1$.

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

Flashcard 27: 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 28: Find the product of $2 + i$ and $2 - i$.

Answer:

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

Flashcard 29: Calculate the discriminant of $x^2 + 4x + 5$.

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

Flashcard 30: What is the degree of the polynomial $0$?

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

Flashcard 31: Find the complex roots of $x^2 + 4$.

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

Flashcard 32: Find the roots of $x^2 - 2x + 1$.

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

Flashcard 33: 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 34: 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 35: What is the degree of $7x^3 + 2x^2 + x + 5$?

Answer:

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

Flashcard 36: 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 37: What is the leading term of $2x^5 + 3x^4$?

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

Flashcard 38: What is a leading coefficient?

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

Flashcard 39: What is the imaginary part of $5 + 6i$?

Answer:

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