Math Core

In Algebra 1 you worked mostly with linear and quadratic expressions. Algebra 2 goes further: cubics, quartics and beyond. These higher-degree polynomials model volumes, revenue curves and the shape of roller-coaster tracks, and every technique in this unit starts with being able to add, subtract and multiply them quickly and accurately.

The language of polynomials

A monomial is a number, a variable, or a product of a number and variables with whole-number exponents, like $7$ , $x$ or $-4x^3$ . A polynomial is a monomial or a sum of monomials, called its terms.

Definition

Degree, leading coefficient, standard form

A polynomial in $x$ is in standard form when its terms are written from the highest power of $x$ down to the lowest. The highest exponent is the degree of the polynomial, and the coefficient of that term is the leading coefficient. The term with no variable is the constant term.

For example, $-2x^4 + 5x^3 - x + 9$ has degree $4$ , leading coefficient $-2$ and constant term $9$ . Notice there is no $x^2$ term: its coefficient is $0$ .

Polynomials are named by degree:

degree	name	example
$0$	constant	$6$
$1$	linear	$3x - 1$
$2$	quadratic	$x^2 + 4x$
$3$	cubic	$2x^3 - x + 5$
$4$	quartic	$x^4 - 16$
$5$	quintic	$-x^5 + x^2$

Expressions like $\dfrac{3}{x}$ or $\sqrt{x}$ are not polynomials, because they would need a negative or fractional exponent on $x$ .

Before you judge the degree, write the polynomial in standard form. In $4x^2 - x^5 + 3$ , the first term is not the leading term: the degree is $5$ and the leading coefficient is $-1$ .

Adding and subtracting

To add polynomials, combine like terms: terms with the same variable raised to the same power. Lining the terms up by degree keeps you organized, especially when some powers are missing.

Subtracting is adding the opposite. Every term of the polynomial being subtracted changes sign.

Worked example: Subtracting cubics

Simplify $(4x^3 - 2x^2 + 7) - (x^3 + 5x^2 - 3x + 2)$ .

Change every sign in the second polynomial and add:

4x^3 - 2x^2 + 7 - x^3 - 5x^2 + 3x - 2.

Now combine like terms, degree by degree:

degree	first	second (negated)	sum
$x^3$	$4$	$-1$	$3$
$x^2$	$-2$	$-5$	$-7$
$x$	$0$	$+3$	$3$
constant	$7$	$-2$	$5$

The result is $3x^3 - 7x^2 + 3x + 5$ .

Common mistake

The minus sign in front of the parentheses applies to every term inside, not just the first. A very common error in the example above is writing $-x^3 + 5x^2 - 3x + 2$ for the second part, which negates only the leading term. When a subtraction has a "missing" term (like the $x$ term above), a sign slip there is easy to overlook.

The sum or difference of two polynomials is always another polynomial. Its degree is at most the larger of the two degrees, and it can be smaller if the leading terms cancel.

Multiplying

To multiply two polynomials, multiply every term of the first by every term of the second, then combine like terms. This is the distributive property used over and over. When you multiply two terms, multiply the coefficients and add the exponents: $(3x^2)(-4x^3) = -12x^5$ .

Degree of a product

If $P$ has degree $m$ and $Q$ has degree $n$ , then $P \cdot Q$ has degree $m + n$ , and its leading coefficient is the product of the two leading coefficients.

This is a fast check on your work: a quadratic times a cubic must be a quintic.

Worked example: Binomial times trinomial

Multiply $(2x - 3)(x^2 + 4x - 5)$ .

Distribute each term of $2x - 3$ :

\begin{aligned} 2x(x^2 + 4x - 5) &= 2x^3 + 8x^2 - 10x \\ -3(x^2 + 4x - 5) &= -3x^2 - 12x + 15 \end{aligned}

Add and combine like terms:

(2x - 3)(x^2 + 4x - 5) = 2x^3 + 5x^2 - 22x + 15.

Check: degree $1 + 2 = 3$ and leading coefficient $2 \cdot 1 = 2$ . ✓ Another quick check is to substitute $x = 1$ into both sides: $(-1)(0) = 0$ and $2 + 5 - 22 + 15 = 0$ . ✓

A table (the "box method") keeps the products organized when both factors have several terms. Put one factor across the top, the other down the side, fill in each product, and then add along the diagonals of like terms.

Special products

A few products appear so often that they're worth knowing by heart:

\begin{aligned} (a + b)^2 &= a^2 + 2ab + b^2 \\ (a - b)^2 &= a^2 - 2ab + b^2 \\ (a + b)(a - b) &= a^2 - b^2 \\ (a + b)^3 &= a^3 + 3a^2 b + 3ab^2 + b^3 \\ (a - b)^3 &= a^3 - 3a^2 b + 3ab^2 - b^3 \end{aligned}

The cube formula comes from multiplying $(a + b)^2 = a^2 + 2ab + b^2$ by one more factor of $a + b$ . The coefficients $1, 3, 3, 1$ are a row of Pascal's triangle, which you'll see again later in the course.

Worked example: Cubing a binomial

Expand $(x + 2)^3$ and $(x - 1)(x + 1)(x^2 + 1)$ .

For the first, use $(a + b)^3$ with $a = x$ and $b = 2$ :

(x + 2)^3 = x^3 + 3(x^2)(2) + 3(x)(2^2) + 2^3 = x^3 + 6x^2 + 12x + 8.

For the second, multiply in a smart order. The first two factors form a difference of squares, and the result pairs with the third factor the same way:

(x - 1)(x + 1)(x^2 + 1) = (x^2 - 1)(x^2 + 1) = x^4 - 1.

Finding one coefficient

Sometimes you only need a single coefficient of a product. You don't have to expand everything: just find every pair of terms whose exponents add up to the power you want.

Worked example: Just the x² coefficient

Find the coefficient of $x^2$ in $(3x^2 - x + 4)(2x^2 + 5x - 6)$ .

The pairs that produce $x^2$ are an $x^2$ term times a constant, an $x$ term times an $x$ term, and a constant times an $x^2$ term:

(3x^2)(-6) + (-x)(5x) + (4)(2x^2) = -18x^2 - 5x^2 + 8x^2 = -15x^2.

The coefficient is $-15$ .

Tip

To check any product, substitute a simple number such as $x = 1$ or $x = 2$ into the original factors and into your answer. If the values differ, there's an error. (Matching values don't prove you're right, but a mismatch always proves you're wrong.)

Practice

Practice 1

Which description fits $5x^2 - 3x^4 + x - 9$ ?

Practice 2

Simplify $(6x^3 + 2x^2 - x + 4) + (-2x^3 + 5x - 7)$ .

Practice 3

Simplify $(3x^2 - 4x + 1) - (5x^2 - 4x - 6)$ .

Practice 4

Multiply $(x + 4)(x^2 - 2x + 3)$ . What is the coefficient of $x$ in the result?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Expand $(x - 3)^3$ .

Practice 6

Without expanding the whole product, find the coefficient of $x^3$ in $(2x^3 - x^2 + 3x - 1)(x^2 + 4x + 5)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

A box has length $2x - 1$ , width $x + 2$ and height $x$ (all in inches). Which polynomial gives its volume?

Practice 8

What is the degree of $(x^3 + 1)^2 (x^2 - 5)^3$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.