02 Polynomial and Rational Functions

A structured guide to recognizing, solving, and graphing polynomial and rational functions using zeros, factors, end behavior, asymptotes, equations, and inequalities.

Recognizing Linear and Quadratic Functions

Functions connect inputs to outputs, and their algebraic forms reveal graph features such as intercepts, zeros, vertices, turning points, and asymptotes. Begin by identifying the function type, then use its structure to choose an appropriate solving or graphing method.

Linear functions

A linear function has the form

f(x)=mx+b.

The value $m$ is the slope, and $b$ is the $y$ -intercept. The graph is a line with $(0,b)$ as its $y$ -intercept. To find its zero, set the function equal to zero:

0=mx+b\quad\Longrightarrow\quad x=-\frac{b}{m},\qquad m\ne0.

For $f(x)=3x-6$ , the slope is $3$ , the $y$ -intercept is $(0,-6)$ , and the zero is $x=2$ , so the graph crosses the $x$ -axis at $(2,0)$ . A constant function such as $f(x)=4$ is a degree-zero polynomial whose graph is horizontal and has no zero.

Quadratic functions

A quadratic function has the form

f(x)=ax^2+bx+c,\qquad a\ne0.

Its graph is a parabola. When $a>0$ , the parabola opens upward and has a minimum; when $a<0$ , it opens downward and has a maximum. Its axis of symmetry is

x=-\frac{b}{2a},

and the vertex is found by evaluating the function at that $x$ -value. The $y$ -intercept is $(0,c)$ .

To solve $ax^2+bx+c=0$ , use factoring, completing the square, or the quadratic formula:

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.

The discriminant is

\Delta=b^2-4ac.

If $\Delta>0$ , there are two distinct real zeros; if $\Delta=0$ , there is one repeated real zero; and if $\Delta<0$ , there are no real zeros.

For example,

x^2-5x+6=(x-2)(x-3),

so the zeros are $x=2$ and $x=3$ .

Takeaway: Identify the function type first. Linear functions use slope and intercept, while quadratic functions use the vertex, axis of symmetry, and discriminant.

Zeros, Factors, and Polynomial Structure

A polynomial function has the form

f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,

where the exponents are nonnegative integers and $a_n\ne0$ . The greatest exponent $n$ is the degree, and $a_n$ is the leading coefficient. Polynomial functions are defined for every real number and have no vertical asymptotes.

Zeros and factors

A zero $r$ satisfies $f(r)=0$ . The Factor Theorem states

f(r)=0\quad\Longleftrightarrow\quad (x-r)\text{ is a factor of }f(x).

A repeated factor gives a zero with greater than one. Odd generally makes the graph cross the $x$ -axis, while even generally makes it touch the axis and turn around. For

f(x)=(x-1)^2(x+2),

the zero $x=1$ has $2$ , so the graph generally touches the axis there. The zero $x=-2$ has $1$ , so the graph generally crosses there.

Possible rational zeros

The limits the rational candidates that need to be tested for a polynomial with integer coefficients. If $\frac{p}{q}$ is a rational zero in lowest terms, then $p$ must divide the constant term and $q$ must divide the leading coefficient.

For

f(x)=2x^3-3x^2-8x+12,

possible candidates include $\pm1$ , $\pm2$ , $\pm3$ , $\pm4$ , $\pm6$ , $\pm12$ , and fractions whose denominators divide $2$ , such as $\pm\frac{1}{2}$ . Testing $x=2$ gives

f(2)=2(8)-3(4)-8(2)+12=0.

Therefore, $x-2$ is a factor, and polynomial division or synthetic division can reduce the remaining problem.

Takeaway: Factoring connects solving and graphing: zeros identify intercepts, and factors reveal how the graph behaves at those intercepts.

Polynomial and Equations

describes the directions a graph takes as $x$ approaches $\infty$ and $-\infty$ . For a polynomial, inspect the degree first and then the sign of the leading coefficient.

An even degree with a positive leading coefficient gives $f(x)\to\infty$ at both ends.
An even degree with a negative leading coefficient gives $f(x)\to-\infty$ at both ends.
An odd degree with a positive leading coefficient gives $f(x)\to-\infty$ as $x\to-\infty$ and $f(x)\to\infty$ as $x\to\infty$ .
An odd degree with a negative leading coefficient gives $f(x)\to\infty$ as $x\to-\infty$ and $f(x)\to-\infty$ as $x\to\infty$ .

For example, $f(x)=-2x^3+4x-1$ has odd degree and a negative leading coefficient. Therefore,

x\to-\infty\Rightarrow f(x)\to\infty, \qquad x\to\infty\Rightarrow f(x)\to-\infty.

A degree- $n$ polynomial has at most $n-1$ turning points and at most $n$ real zeros.

Solving polynomial equations

Use this sequence:

Move every term to one side so the equation equals zero.
Factor as completely as possible.
Apply the Zero Product Property: if $AB=0$ , then $A=0$ or $B=0$ .
Use the quadratic formula on any remaining quadratic factor.
Check the solutions in the original equation.

For example,

x^3-4x^2-x+4=0

can be grouped and factored as

x^2(x-4)-1(x-4)=0,

(x^2-1)(x-4)=0,

(x-1)(x+1)(x-4)=0.

Thus, the solutions are $x=1$ , $x=-1$ , and $x=4$ .

Takeaway: Degree and leading coefficient predict distant graph behavior, while factorization identifies real zeros and helps reveal turning behavior.

Sign Charts and Rational-Function Structure

A polynomial inequality is solved by determining where a factored expression is positive, negative, zero, or undefined. A makes the interval analysis systematic.

Polynomial inequalities

Move all terms to one side.
Factor the resulting expression.
Find its real zeros.
Place those critical numbers on a number line.
Test one value in each interval.
Include zeros for $\le$ or $\ge$ , but exclude them for $<$ or $>$ .

For

(x-2)(x+1)\ge0,

the critical numbers are $x=-1$ and $x=2$ . The expression is positive on $(-\infty,-1)$ and $(2,\infty)$ , and negative on $(-1,2)$ . Because the inequality is nonnegative, the endpoints are included:

(-\infty,-1]\cup[2,\infty).

Rational functions

A rational function is a quotient of polynomial functions:

f(x)=\frac{p(x)}{q(x)},\qquad q(x)\ne0.

Its domain excludes every value that makes the original denominator zero. Always factor the numerator and denominator before interpreting the graph.

A common factor that cancels creates a , or hole.
A denominator zero that remains after cancellation creates a .

For

f(x)=\frac{(x-2)(x+1)}{(x-2)(x-3)},

simplification gives

f(x)=\frac{x+1}{x-3},\qquad x\ne2.

The canceled factor creates a hole at $x=2$ . Its missing output is found using the simplified function:

y=\frac{2+1}{2-3}=-3.

Thus, the hole is $(2,-3)$ , while the remaining denominator gives a at $x=3$ .

Takeaway: In inequalities, zeros divide the number line into sign intervals. In rational functions, canceled denominator factors create holes and uncanceled denominator factors create vertical asymptotes.

Rational Asymptotes and Graphing

Rational-function behavior far from the origin is determined by comparing the degrees of the numerator and denominator after common factors have been canceled.

Horizontal and slant behavior

A describes the value approached as $x$ becomes very large in either direction.

If the numerator degree is less than the denominator degree, the is $y=0$ .
If the degrees are equal, the is the ratio of the leading coefficients.
If the numerator degree is greater, there is no .

For

f(x)=\frac{3x^2+1}{x^3-4},

the numerator degree is smaller, so the is $y=0$ . For

g(x)=\frac{4x^3-x}{2x^3+5},

the degrees are equal, so the is

y=\frac{4}{2}=2.

When the numerator degree is exactly one greater than the denominator degree, division gives a . For

h(x)=\frac{x^2+1}{x-1},

division gives

h(x)=x+1+\frac{2}{x-1},

so the is $y=x+1$ , and the is $x=1$ .

A graphing sequence

To sketch a rational function:

Factor the numerator and denominator.
State the domain restrictions from the original denominator.
Identify holes from canceled factors.
Identify vertical asymptotes from uncanceled denominator zeros.
Find zeros from uncanceled numerator zeros.
Evaluate $f(0)$ for the $y$ -intercept when it is defined.
Compare degrees to find a horizontal or .
Test points in intervals separated by zeros, holes, and vertical asymptotes.
Determine the behavior near each asymptote and at both ends.

A rational graph may cross a horizontal or , but it cannot cross a at a point in its domain.

Takeaway: Cancellation determines holes, denominator zeros determine vertical asymptotes, and degree comparison determines the graph's distant trend.

Solving Rational Equations and Inequalities

Rational equations and inequalities require careful attention to values that make an original denominator zero. Such values remain excluded even if algebraic manipulation appears to remove them.

Rational equations

Use the following procedure:

Set every original denominator equal to zero and record the restrictions.
Multiply both sides by the least common denominator.
Solve the resulting equation.
Reject any solution that violates an original restriction.
Check each remaining solution in the original equation.

For

\frac{2}{x-1}=\frac{3}{x+2},

the restrictions are $x\ne1$ and $x\ne-2$ . Multiplying by $(x-1)(x+2)$ gives

2(x+2)=3(x-1).

Solving produces

2x+4=3x-3\quad\Longrightarrow\quad x=7.

Because $7$ satisfies the restrictions, it is the solution.

Rational inequalities

Move all terms to one side.
Combine the result into one rational expression.
Factor the numerator and denominator.
Find critical numbers from numerator zeros and denominator zeros.
Test the intervals in a .
Include numerator zeros for a non-strict inequality.
Never include denominator zeros, because the expression is undefined there.

For

\frac{x-2}{x+1}\le0,

the critical numbers are $x=2$ , from the numerator, and $x=-1$ , from the denominator. The expression is negative on $(-1,2)$ and positive on the other intervals. Include $2$ , but exclude $-1$ , giving

(-1,2].

Final checklist: Factor before interpreting, preserve all original restrictions, use zeros and denominator exclusions as interval boundaries, and verify solutions in the original equation or inequality.

Takeaway: Algebraic simplification can reveal structure, but it never restores a value excluded by the original domain.