Precalculus Polynomial and Rational Functions Syllabus

For $f(x) = 2x^{3} + x - 6$, list the possible rational zeros from the Rational Zero Theorem.

$p$ divides 6: $\pm1, \pm2, \pm3, \pm6$; $q$ divides 2: $\pm1, \pm2$. Possible zeros: $\pm1, \pm2, \pm3, \pm6, \pm\tfrac{1}{2}, \pm\tfrac{3}{2}$.

State the Fundamental Theorem of Algebra.

Every polynomial of degree $n \geq 1$ with complex coefficients has at least one complex zero.

How many zeros (counting multiplicity) does a degree-$n$ polynomial have according to the Linear Factorization Theorem?

Exactly $n$ complex zeros, counting multiplicities; it factors completely into $n$ linear factors over the complex numbers.

State Descartes's Rule of Signs for the number of positive real zeros.

The number of positive real zeros of $f(x)$ equals the number of sign changes in $f(x)$, or is less than that by an even number.

Using Descartes's Rule of Signs, how do you find the possible number of negative real zeros?

Count the sign changes in $f(-x)$; the number of negative real zeros equals that count or is less by an even number.

State the Conjugate Pairs (Complex Conjugate Zeros) Theorem.

If $f(x)$ has real coefficients and $a + bi$ (with $b \neq 0$) is a zero, then its conjugate $a - bi$ is also a zero.

Why must complex zeros of a real-coefficient polynomial occur in pairs?

Because of the Conjugate Pairs Theorem, non-real zeros come in conjugate pairs; hence a real-coefficient polynomial of odd degree must have at least one real zero.

Define the imaginary unit $i$.

$i = \sqrt{-1}$, so $i^{2} = -1$.

What is the standard form of a complex number?

$a + bi$, where $a$ is the real part and $b$ is the imaginary part, and $a, b$ are real numbers.

Simplify the powers $i^{1}, i^{2}, i^{3}, i^{4}$.

$i^{1} = i$, $i^{2} = -1$, $i^{3} = -i$, $i^{4} = 1$. The pattern repeats with period 4.

How do you add two complex numbers $(a + bi) + (c + di)$?

$(a + c) + (b + d)i$ — add real parts and imaginary parts separately.

How do you multiply two complex numbers $(a + bi)(c + di)$?

Use FOIL and $i^{2} = -1$: $(ac - bd) + (ad + bc)i$.

What is the complex conjugate of $a + bi$, and what is the product of a number with its conjugate?

The conjugate is $a - bi$. Their product is $(a + bi)(a - bi) = a^{2} + b^{2}$, a nonnegative real number.

How do you divide complex numbers, e.g. $\dfrac{a + bi}{c + di}$?

Multiply numerator and denominator by the conjugate of the denominator: $\dfrac{(a+bi)(c-di)}{c^{2}+d^{2}}$, then write in standard form.

Write $\sqrt{-9}$ in terms of $i$.

$\sqrt{-9} = 3i$, since $\sqrt{-9} = \sqrt{9}\,\sqrt{-1} = 3i$.

How is the domain of a rational function determined?

All real numbers except the values of $x$ that make the denominator equal to zero (excluded values).

What distinguishes a vertical asymptote from a removable discontinuity (hole) in a rational function?

A factor causing zero in the denominator that does NOT cancel gives a vertical asymptote; a factor that cancels with the numerator gives a removable discontinuity (hole).

How do you locate the vertical asymptotes of a rational function in lowest terms?

Set the (simplified) denominator equal to zero and solve; each such $x$-value where the numerator is nonzero gives a vertical asymptote.

State the rule for the horizontal asymptote when the degree of the numerator is LESS than the degree of the denominator.

The horizontal asymptote is $y = 0$ (the $x$-axis).

State the rule for the horizontal asymptote when the numerator and denominator have EQUAL degree.

The horizontal asymptote is $y = \dfrac{a_{n}}{b_{m}}$, the ratio of the leading coefficients.

What happens to horizontal asymptotes when the degree of the numerator is GREATER than the degree of the denominator?

There is no horizontal asymptote. If the numerator's degree is exactly one more than the denominator's, there is a slant (oblique) asymptote instead.

How do you find the slant (oblique) asymptote of a rational function?

Perform polynomial long division; the quotient (ignoring the remainder) gives the line $y = mx + b$ that is the slant asymptote.

Can the graph of a rational function cross its horizontal asymptote?

Yes, a graph may cross a horizontal (or slant) asymptote in the middle of the graph, but it cannot cross a vertical asymptote.

How do you find the $x$-intercepts of a rational function?

Set the numerator (in lowest terms) equal to zero and solve; these are the values where $f(x) = 0$ and the denominator is nonzero.

How do you convert a quadratic from standard form to vertex form?

By completing the square: factor $a$ from the $x$-terms, add and subtract $\left(\dfrac{b}{2a}\right)^{2}$ inside, and rewrite as $a(x-h)^{2}+k$.

Factor completely and give all zeros of $f(x) = x^{2} + 4$ over the complex numbers.

$f(x) = (x - 2i)(x + 2i)$; the zeros are $x = 2i$ and $x = -2i$.

If a cubic polynomial with real coefficients has $2 + i$ as a zero, name another zero it must have.

$2 - i$ (its complex conjugate), by the Conjugate Pairs Theorem.

What is the remainder when $f(x) = x^{3} - 2x + 5$ is divided by $x - 2$?

By the Remainder Theorem, $f(2) = 8 - 4 + 5 = 9$.

How does the multiplicity of a zero relate to the graph's flatness at the intercept?

The higher the multiplicity, the flatter the graph becomes as it touches or crosses the $x$-axis at that zero (it flattens near the intercept).

Compare the end behavior of $f(x) = -3x^{4} + \cdots$ and $g(x) = 5x^{3} + \cdots$.

For $f$ (even degree, $a<0$): both ends fall, $f(x)\to-\infty$ as $x\to\pm\infty$. For $g$ (odd degree, $a>0$): falls left, rises right ($g\to-\infty$ as $x\to-\infty$, $g\to+\infty$ as $x\to+\infty$).