Polynomials and Their Graphs: Practice

A polynomial is a finite sum of terms $c\,x^k$ where each exponent $k$ is a whole number. Dividing by $x$, taking a root of $x$, or putting $x$ in an exponent all break that pattern.

$4x^3 - x + 2 = 4x^3 + (-1)x^1 + 2x^0$

Every term has a real coefficient and a whole-number exponent, so $4x^3 - x + 2$ is a polynomial. The other three divide by the variable, take a root of it, or place it in the exponent.

The degree is the highest power of $x$ with a nonzero coefficient. The polynomial is already in standard form, so read the first exponent.

$7x^5 - 3x^2 + x - 8 \quad\Rightarrow\quad \text{degree } 5$

The $7$ is the leading coefficient, not the degree, and $-8$ is the constant term.

The polynomial is in standard form, so the leading term is the first one, $4x^3$.

$\text{leading term } 4x^3 \quad\Rightarrow\quad \text{leading coefficient } 4$

The number $3$ is the degree, $-9$ is the coefficient of $x^2$, and $6$ is the constant term.

Count the terms after combining like terms: $4x^2$, $-x$, and $6$.

$4x^2 - x + 6 \quad\Rightarrow\quad 3 \text{ terms}$

Three terms make a trinomial (one term is a monomial, two make a binomial).

Wrap the negative input in parentheses before taking the power. An odd power keeps the sign of its base.

$g(-1) = 2(-1)^3 = 2(-1) = -2$

The cube of $-1$ is $-1$, so the output is $-2$.

A nonzero constant is the term $12x^0$, so its highest power of $x$ is the zeroth.

$p(x) = 12 = 12x^0 \quad\Rightarrow\quad \text{degree } 0$

Only the zero polynomial $f(x) = 0$ is assigned no degree, because it has no nonzero term at all.

A polynomial of degree $n$ has at most $n$ real zeros.

$n = 6 \quad\Rightarrow\quad \text{at most } 6 \text{ real zeros}$

The ceiling equals the degree itself; it may have fewer, but never more.

The leading term is $x^2$, with an even degree and a positive leading coefficient.

$n = 2 \text{ (even)}, \qquad a_n = 1 > 0$

An even degree makes the two ends agree, and the positive coefficient points them up: both ends rise, the familiar upward parabola.

Polynomial terms have the form $c\,x^k$ with $k$ a whole number: $0, 1, 2, 3, \ldots$

$k = -2 \quad\Rightarrow\quad c\,x^{-2} = \frac{c}{x^2}$

A negative exponent means dividing by the variable, which polynomials never do. Exponents $0$, $3$, and $10$ are all legal.

The $y$-intercept is the output at $x = 0$, and evaluating at zero kills every term that carries an $x$.

$f(0) = 0 - 0 + 9 = 9$

So the graph crosses the $y$-axis at $(0, 9)$: the constant term is the $y$-intercept height.

Evaluating a polynomial uses only powers, multiplications by constants, and additions, and every real number can be raised to a whole-number power, scaled, and added.

$\text{domain} = \text{all real numbers}$

There is no division by the variable and no root of it, so no input ever has to be excluded.

Polynomial outputs change gradually as inputs change gradually, and every real number is a legal input.

$\text{polynomial graph} \;\Rightarrow\; \text{no gaps, no jumps, no corners}$

So the graph is always a smooth, unbroken curve. Corners belong to graphs like $y = |x|$ and breaks to graphs like $y = \dfrac{1}{x}$, neither of which is a polynomial.