Polynomial Functions and Complex Zeros - AP Precalculus

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

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

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

1. The highest power term determines the degree.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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