2 - Polynomial Functions

A structured guide to polynomial functions, their algebraic operations, zeros, division methods, graph behavior, and complex roots.

Polynomial Operations and Structure

A is a finite sum of terms whose variable exponents are nonnegative integers:

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

with $a_n\neq 0$ . The is the greatest exponent, and the is the coefficient of the highest- term. For example, $4x^3-2x+7$ has $3$ and $4$ .

Polynomial functions are continuous: their graphs have no breaks, holes, or vertical asymptotes. They can model quantities such as area, volume, population trends, and approximations.

Performing operations

To add or subtract polynomials, combine like terms, which have the same variable and exponent.
To multiply, distribute every term in one polynomial across every term in the other.
To evaluate, substitute a value for the variable. If $f(x)=2x^3-x^2+4x-1$ , then $f(2)=27$ .
The of a product is the sum of the degrees of its nonzero factors.
The of a sum is at most the greatest among its terms; cancellation can make it smaller.

Useful identities include

(a+b)^2=a^2+2ab+b^2,

(a-b)^2=a^2-2ab+b^2,

and

(a+b)(a-b)=a^2-b^2.

Takeaway: Organize terms by exponent, apply the distributive property carefully, and identify and before analyzing a polynomial’s graph.

Zeros, Factors, and

A of $f$ is a number $c$ for which $f(c)=0$ . On a graph, a real is an $x$ -intercept. The gives the exact connection between zeros and factors:

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

For instance, if $f(3)=0$ , then $x-3$ is a factor of $f(x)$ .

The of a is the number of times its factor occurs. In

f(x)=(x-2)^3(x+1)^2,

the $x=2$ has $3$ , while $x=-1$ has $2$ .

predicts graph behavior:

An odd usually means the graph crosses the $x$ -axis.
An even means the graph touches the $x$ -axis and turns around.
A greater than $1$ generally makes the graph flatter near the intercept.

For example, $y=(x-1)^2$ touches the axis at $x=1$ , whereas $y=(x-1)^3$ crosses it with a flattened shape.

Takeaway: Factoring reveals zeros, and tells you how the graph behaves at each real .

Polynomial Division and Remainders

Polynomial division rewrites a dividend as a product plus a remainder:

f(x)=d(x)q(x)+r(x),

where $r(x)=0$ or the of $r(x)$ is less than the of $d(x)$ .

In long division, repeatedly divide the leading term, multiply, subtract, and bring down the next term. For example,

2x^3+3x^2-5x+6=(x+2)(2x^2-x-3)+12.

Therefore,

\frac{2x^3+3x^2-5x+6}{x+2}=2x^2-x-3+\frac{12}{x+2}.

is a shorter method for divisors of the form $x-c$ . When dividing by $x+2=x-(-2)$ , use $-2$ in the synthetic-division setup. The quotient is $2x^2-x-3$ , and the remainder is $12$ .

The provides a faster way to find that remainder:

f(x)=(x-c)q(x)+f(c).

For $f(x)=x^3-4x+1$ , division by $x-2$ has remainder

f(2)=2^3-4(2)+1=1.

Because the remainder is not , $x-2$ is not a factor.

Takeaway: A remainder confirms a factor, while a nonzero remainder shows that the proposed linear factor does not divide evenly.

Finding Polynomial Zeros

To find polynomial zeros systematically, first write the polynomial in standard form and factor any common factors or recognizable patterns. If the coefficients are integers, use the to create a list of possible rational zeros. Test candidates by substitution or , then continue factoring the quotient.

For

f(x)=x^3-6x^2+11x-6,

possible rational zeros include $\pm1$ , $\pm2$ , $\pm3$ , and $\pm6$ . Testing $x=1$ gives , so $x-1$ is a factor. gives

f(x)=(x-1)(x^2-5x+6).

Factoring the quadratic produces

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

Thus the zeros are

x=1,\ 2,\ 3.

The identifies candidates only; it does not guarantee that every candidate is a . If a quadratic remains after factoring, solve it by factoring or by using the quadratic formula.

Takeaway: Use the theorem to reduce the search, verify candidates, and repeat division and factoring until all factors are resolved.

and Graphing

is determined by the leading term. For a leading term $a_nx^n$ , inspect whether the is even or odd and whether the is positive or negative:

Even with positive : both ends rise.
Even with negative : both ends fall.
Odd with positive : the left end falls and the right end rises.
Odd with negative : the left end rises and the right end falls.

For example, $f(x)=3x^4-2x+1$ has even and a positive , so both ends rise. In contrast, $g(x)=-2x^5+x^2-7$ has odd and a negative , so the left end rises and the right end falls.

In limit notation,

\lim_{x\to\infty}3x^4=\infty, \qquad \lim_{x\to-\infty}3x^4=\infty,

while

\lim_{x\to\infty}(-2x^5)=-\infty, \qquad \lim_{x\to-\infty}(-2x^5)=\infty.

To graph a polynomial, determine the and , establish , find real zeros and their multiplicities, plot the $y$ -intercept $f(0)$ , and draw a smooth curve consistent with these facts. A - $n$ polynomial has at most $n-1$ turning points.

Takeaway: Start the graph at both far ends using the leading term, then use intercepts and multiplicities to shape the middle.

Complex Zeros and Complete Factorization

The guarantees that every nonconstant polynomial with complex coefficients has at least one complex . More generally, a - $n$ polynomial has exactly $n$ complex zeros when multiplicities are counted. It can therefore be written as

f(x)=a_n(x-c_1)(x-c_2)\cdots(x-c_n),

where the zeros may be real or nonreal complex numbers.

For example, $x^2+4$ has no real zeros, but

x^2+4=0

gives

x=\pm2i.

When the coefficients are real, occur in pairs. If $2+3i$ is a , then $2-3i$ is also a . Their corresponding factors combine to form a real quadratic:

\begin{aligned} (x-(2+3i))(x-(2-3i)) &=((x-2)-3i)((x-2)+3i)\\ &=(x-2)^2+9. \end{aligned}

Thus, a polynomial may have fewer real intercepts than its while still having the full number of complex zeros required by the theorem.

Takeaway: Count real and nonreal zeros together, including ; nonreal zeros of real-coefficient polynomials appear in conjugate pairs.