Polynomials in One Variable

A polynomial is the most well-behaved algebraic expression there is. You build one with three ingredients: a variable, real-number coefficients, and the operations of addition and multiplication (with whole-number powers of the variable). No division by the variable, no square roots of the variable, no $\sin x$ , those are forbidden. The result is an expression simple enough to compute with by hand yet rich enough to model real curves.

Definitions

A polynomial in one variable $x$ is an expression of the form $p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0,$ where $n$ is a non-negative integer and $a_0, a_1, \ldots, a_n$ are real numbers called coefficients. If $a_n \ne 0$, the highest power $n$ is called the degree of the polynomial, and $a_n$ is its leading coefficient.

The constant $a_0$ is the constant term. A non-zero constant like $7$ is a polynomial of degree $0$. The expression $0$ alone is the zero polynomial, whose degree is, by convention, undefined (or sometimes called $-\infty$).

A monomial is a polynomial with one term, like $5x^3$. A binomial has two terms, like $2x + 1$. A trinomial has three terms, like $x^2 - 5x + 6$.

Concept and types

By degree, polynomials get the following names:

• Degree $0$: constant polynomial. Example: $p(x) = 7$.
• Degree $1$: linear polynomial. Example: $p(x) = 3x - 2$.
• Degree $2$: quadratic polynomial. Example: $p(x) = x^2 - 5x + 6$.
• Degree $3$: cubic polynomial. Example: $p(x) = 2x^3 - x + 4$.
• Degree $4$: quartic; degree $5$: quintic; and so on.

Standard form. A polynomial is in standard form when its terms are written in decreasing order of powers, with no missing-power gaps written explicitly. So $3 - 5x + 2x^2$ in standard form is $2x^2 - 5x + 3$.

Operations. Polynomials can be added, subtracted, and multiplied. Add and subtract by combining like terms (same power of $x$). Multiply by distributing every term of one polynomial with every term of the other: $(2x + 3)(x - 4) = 2x \cdot x + 2x \cdot (-4) + 3 \cdot x + 3 \cdot (-4) = 2x^2 - 5x - 12.$

Evaluation. To evaluate a polynomial at $x = c$, substitute $c$ in place of $x$ and simplify. If $p(x) = x^2 - 3x + 4$, then $p(2) = 4 - 6 + 4 = 2$.

What is not a polynomial? Several common expressions fail the test:

• $\dfrac{1}{x} = x^{-1}$: the power is negative.
• $\sqrt{x} = x^{1/2}$: the power is fractional.
• $\dfrac{x^2 + 1}{x - 1}$: an entire polynomial divided by another polynomial is generally a rational expression, not a polynomial.
• $\sin x, e^x, \log x$: these are not built from $x$ using only $+$, $-$, $\times$ and whole-number powers.

But $\sqrt{2}\,x^2 + 3x$ is a polynomial: the coefficient $\sqrt{2}$ is a real number, even though it is irrational, and the powers of $x$ are whole numbers ($2$ and $1$).

Polynomials in more than one variable exist too (e.g. $x^2 + 2xy + y^2$), but in this chapter we focus on one variable.

Worked examples

Example 1. State the degree, leading coefficient and constant term of $p(x) = 4x^3 - 7x + 2$.

Degree: $3$. Leading coefficient: $4$. Constant term: $2$.

Example 2. Add $p(x) = x^2 + 3x - 5$ and $q(x) = 2x^2 - x + 7$.

Combine like terms: $(1+2)x^2 + (3-1)x + (-5+7) = 3x^2 + 2x + 2$.

Example 3. Multiply $(x + 2)(x^2 - x + 4)$.

Distribute: $x(x^2 - x + 4) + 2(x^2 - x + 4) = x^3 - x^2 + 4x + 2x^2 - 2x + 8 = x^3 + x^2 + 2x + 8$.

Example 4. Evaluate $p(x) = 2x^3 - 5x + 1$ at $x = -1$.

$p(-1) = 2(-1)^3 - 5(-1) + 1 = -2 + 5 + 1 = 4$.

Example 5. Which of the following are polynomials? $\dfrac{1}{x} + 3$, $\sqrt{2}\,x + 1$, $x^2 - \sqrt{x}$, $3x^4 - 2x^2 + 7$.

Polynomial: $\sqrt{2}x + 1$ (irrational coefficient is fine, power is $1$), and $3x^4 - 2x^2 + 7$ (whole-number powers and real coefficients). Not polynomials: $\dfrac{1}{x} + 3$ (negative power) and $x^2 - \sqrt{x}$ (fractional power).

Try it yourself

1. Write the degree of each: $7$, $3x - 2$, $-x^2 + 5$, $4x^3 - x + 1$, $x^7$.
2. State the leading coefficient and constant term of $2x^4 - 3x^2 + 7$.
3. Which of the following are polynomials? $3x + 4$, $\dfrac{1}{x} + 1$, $5\sqrt{x}$, $x^{1/2} + 2$, $7$.
4. Add $3x^2 + 2x - 1$ and $-x^2 + 4x + 5$.
5. Multiply $(x - 3)(x + 5)$.
6. Multiply $(2x + 1)(x^2 - x + 3)$.
7. If $p(x) = x^3 - 4x^2 + x - 2$, find $p(0), p(1), p(-2)$.
8. Write a quadratic polynomial whose constant term is $-6$ and whose leading coefficient is $1$.
9. Classify by number of terms: $5x^3$, $x^2 + 3$, $4x - 7$, $x^3 + x^2 + x$.
10. Find a binomial that is a cubic polynomial.

Pitfalls / Insight

• Coefficients can be irrational. $\sqrt{2}\, x + 3$ is a polynomial; the restriction is on the power, not the coefficient.
• Negative or fractional powers disqualify the expression. $\dfrac{1}{x}$ and $\sqrt{x}$ are not polynomials.
• The zero polynomial is special. It has no defined degree.

Insight. Polynomials are addition-and-multiplication-only objects: nothing exotic, nothing tricky. That very plainness is why they are everywhere , once you can compute with them, you can approximate almost any smooth function and any continuous curve.