Adding and Subtracting Polynomials

Learning goals

Name a polynomial's terms, degree and leading coefficient
Write standard form in descending order of degree
Combine only like terms, those sharing a power
Distribute the minus across every term when subtracting
Predict the degree of a sum, allowing for cancellation

What a polynomial is

A polynomial in the variable $x$ is a finite sum of terms, where each term is a real number times a whole-number power of $x$ :

a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0.

The numbers $a_n, \ldots, a_1, a_0$ are the coefficients, and every exponent must be a nonnegative integer ( $0, 1, 2, \ldots$ ). That requirement is exactly what separates a polynomial from other expressions. No variable sits in a denominator, no variable hides under a root, and no power is negative or fractional. There are only whole-number powers of $x$ , each scaled by a coefficient and added together.

Reading a concrete polynomial term by term pins down the vocabulary you will use for the rest of the chapter. Take

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

Term	Coefficient	Degree
$3x^4$	$3$	$4$
$-5x^2$	$-5$	$2$
$7x$	$7$	$1$
$-2$	$-2$	$0$

The degree of a term is the power of $x$ in it. The term $7x$ is really $7x^1$ , so its degree is $1$ , and the constant $-2$ is really $-2x^0$ (since $x^0 = 1$ ), so its degree is $0$ . The degree of the polynomial is the largest degree among its terms, which is $4$ here. The highest-degree term, $3x^4$ , is the leading term, and its coefficient $3$ is the leading coefficient. The constant term is the term with no variable, the $x^0$ term, which is $-2$ . Notice that this polynomial has no $x^3$ term. That simply means its $x^3$ coefficient is $0$ , and a term with a zero coefficient contributes nothing and is left out.

Short polynomials get their own names by their number of terms. A one-term polynomial such as $7x$ or $4x^3$ or $9$ is a monomial. A two-term polynomial such as $x + 1$ is a binomial. A three-term polynomial such as $x^2 - 5x + 6$ is a trinomial. Past three terms there is no special word, and you simply call it a polynomial.

Standard form

A polynomial is in standard form when its terms are written in descending order of degree, the highest power first and the constant last. The polynomial $3x^4 - 5x^2 + 7x - 2$ above is already in standard form. You could rearrange it as $7x - 2 + 3x^4 - 5x^2$ , and it would equal the same number for every value of $x$ , but that scrambled order hides everything useful about it.

Descending order earns its place as the default for three reasons. The degree and the leading term sit right at the front, so you can read a polynomial’s most important features at a glance. Two polynomials written in standard form can be compared term by term, the way you compare two whole numbers digit by digit. And when you add or subtract, lining terms up by descending power drops like terms into the same column automatically. That alignment is exactly the setup the rest of this lesson uses.

Check your understanding

For the polynomial $6x^3 - x^5 + 4 - 2x$ , what is the leading coefficient?

6

-1

4

1

Why only like terms combine

Like terms are terms with the same variable part, meaning the same power of $x$ . In a single variable that just means the same exponent. So $3x^2$ and $-8x^2$ are like terms, and $5x$ and $x$ are like terms, but $x^2$ and $x^3$ are not, and neither are $4x$ and $4$ . Only like terms can be merged into a single term, and the reason is the distributive property, run backward.

Why like terms combine and unlike terms do not#

The distributive property says that for any numbers and quantities, $ba + ca = (b + c)a$ . Let the shared quantity $a$ be a single power of $x$ , say $x^2$ . Two terms that both carry that power are $b$ copies and $c$ copies of the very same thing, so

3x^2 + 5x^2 = (3 + 5)x^2 = 8x^2.

The common factor $x^2$ pulls out to the right, the coefficients $3$ and $5$ add, and the power itself is untouched. That is the whole mechanism: combining like terms is the distributive property read from left to right. That combination works only because both terms are copies of the identical power. The exponent stays $2$ , so the answer is $8x^2$ , never $8x^4$ .

Now try two different powers, $x^2$ and $x^3$ . Factoring the smaller power out is the most you can do,

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

and $1 + x$ is not a number, so the result is not a single term $c\,x^k$ for any constant $c$ and power $k$ . There is nothing to add, because the two terms are not copies of one shared power. It helps to remember what those powers measure: $x^2$ is an area and $x^3$ is a volume. Adding an area to a volume collapses to a single number no more than adding $3$ metres to $5$ square metres does. Like terms carry the same power and merge into one; unlike terms carry different powers and stay side by side.

The rule to carry forward is short: to combine like terms, add their coefficients and keep the shared power unchanged.

Adding polynomials

Adding two polynomials is now almost mechanical. A plus sign in front of a parenthesis changes nothing inside it, so you can erase the parentheses, gather the like terms, and add their coefficients.

Worked example 1 Add $(3x^2 + 5x - 2) + (x^2 - 4x + 6)$

The leading plus sign lets the parentheses drop with every sign kept as it stands:

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

Group the like terms by power, then add the coefficients in each group:

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

The $x^2$ terms give $3 + 1 = 4$ , the $x$ terms give $5 + (-4) = 1$ , and the constants give $-2 + 6 = 4$ . The sum $4x^2 + x + 4$ is already in standard form.

Lining the terms up in columns, one column per power, makes the bookkeeping visible. It is ordinary column addition, with powers of $x$ playing the role that place values play for whole numbers.

Adding polynomials by lining up like terms in columns, one column per power of x, then adding down each column. It is column addition, with powers of x in place of place values.

When one polynomial is missing a power the other one has, leave that column empty rather than forcing an entry. The gap just means the coefficient there is $0$ .

Worked example 2 Add $(2x^3 + x - 5) + (4x^2 - 3x + 1)$

The first polynomial has no $x^2$ term and the second has no $x^3$ term, so those powers each appear in only one of the two. Drop the parentheses and group by power:

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

Only the $x$ terms and the constants have partners to combine:

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

The $2x^3$ and the $4x^2$ carry straight down, because nothing is there to add to them. Writing the result in descending order gives $2x^3 + 4x^2 - 2x - 4$ .

Subtracting polynomials

Subtraction is where nearly every mistake happens, and it has a single cause. Subtracting a polynomial means subtracting every term in it, not just the first. A minus sign in front of a parenthesis is a factor of $-1$ multiplying the whole group. By the distributive property, that factor flips the sign of each term inside:

-(2x^2 + 6x - 7) = -2x^2 - 6x + 7.

The $2x^2$ becomes $-2x^2$ , the $+6x$ becomes $-6x$ , and the $-7$ becomes $+7$ . Every sign turns over. Once the parentheses are gone, you combine like terms exactly as with addition.

Worked example 3 Subtract $(5x^2 - 3x + 4) - (2x^2 + 6x - 7)$

Distribute the minus sign across all three terms of the second polynomial, flipping each sign:

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

Watch the two signs that changed: $+6x$ became $-6x$ , and $-7$ became $+7$ . Now group and combine:

(5x^2 - 2x^2) + (-3x - 6x) + (4 + 7) = 3x^2 - 9x + 11.

Had you flipped only the first term and written $5x^2 - 3x + 4 - 2x^2 + 6x - 7$ , the $x$ term and the constant would both come out wrong. The sign change has to reach every term.

Worked example 4 Subtract $(4x^3 - 2x^2 + x) - (x^3 - 5x^2 - 3x + 2)$

The second polynomial has four terms, so four signs flip. In particular the $-5x^2$ becomes $+5x^2$ , and the $+2$ becomes $-2$ :

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

Group by power and add the coefficients in each group:

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

The middle group is the one to double-check: $-2x^2$ plus the flipped $+5x^2$ gives $+3x^2$ , not $-7x^2$ . Subtracting a negative term adds it back, which is precisely what the distributed minus sign takes care of.

Check your understanding

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

3x^2 + x - 2

3x^2 - 3x + 8

5x^2 - 3x + 8

3x^2 - 3x - 2

Closure and the degree of a sum

Add or subtract two polynomials and you always get another polynomial. Adding or subtracting real coefficients gives real coefficients, and the powers involved were already nonnegative integers, so the result still fits the definition. Mathematicians describe this by saying the polynomials are closed under addition and subtraction, the same way the whole numbers are closed under addition.

The degree of the result is usually easy to predict. When you add two polynomials, the highest power present in either one is still present in the sum. Because that power is still present, the degree of the sum is the larger of the two degrees. There is one exception worth knowing, and it happens only when the two polynomials share the same degree and their leading terms cancel.

Worked example 5 Compare the degree of $(x^2 + x) + (x^2 - 1)$ with $(x^2 + x) - (x^2 - 1)$

Both polynomials have degree $2$ , with the same leading term $x^2$ . Add them first:

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

The leading coefficients are $1$ and $1$ , and $1 + 1 = 2$ is not zero, so the $x^2$ term survives and the sum keeps degree $2$ . Now subtract instead:

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

This time the $x^2$ terms cancel, since $1 - 1 = 0$ , and the highest power left is $x^1$ . The difference has degree $1$ , lower than either polynomial you started with. The same two polynomials give a degree- $2$ sum and a degree- $1$ difference. The difference comes out lower because subtraction lined up equal leading terms with opposite signs, and they wiped each other out.

Check your understanding

What is the degree of $(3x^2 + 2x - 1) - (3x^2 - x + 4)$ ?

2

1

0

3

Common mistakes

Dropping the negative on the later terms of a subtraction. A minus sign in front of a parenthesis multiplies every term inside by $-1$ . In $(5x^2 - 3x + 4) - (2x^2 + 6x - 7)$ the $+6x$ must become $-6x$ and the $-7$ must become $+7$ . Flipping only the first term is the single most common error in the whole chapter.
Combining unlike terms. Only terms with the same power merge. $3x^2 + 4x$ stays $3x^2 + 4x$ ; it is not $7x^2$ and not $7x^3$ . If the powers differ, leave the terms side by side.
Changing the exponent when combining like terms. Adding like terms adds the coefficients and keeps the power: $3x^2 + 5x^2 = 8x^2$ , never $8x^4$ . You are counting copies of one power, not multiplying powers.
Losing the constant term. The constant is the $x^0$ term, and it combines only with other constants. It is easy to carry down a row of variable terms and forget the plain number at the end. The mistake is especially easy in a subtraction, where the constant’s sign has just flipped.
Mislabeling the leading coefficient. The leading coefficient belongs to the highest-degree term, which is not always the first one written. In $6x^3 - x^5 + 4$ the leading term is $-x^5$ , so the leading coefficient is $-1$ . Put the polynomial in standard form first.

Key takeaways

A polynomial is a finite sum of terms, each a real coefficient times a nonnegative-integer power of $x$ . Key words: term, coefficient, degree of a term and of the polynomial, leading term and leading coefficient, and constant term. The words monomial / binomial / trinomial name polynomials of one, two, and three terms.
Standard form writes the terms in descending order of degree, so the degree and leading term are visible up front and like terms line up when you add.
Like terms carry the same power. Combine them by adding coefficients and keeping the power unchanged, because $3x^2 + 5x^2 = (3 + 5)x^2 = 8x^2$ is the distributive property run backward. Unlike terms cannot merge.
To add, drop the parentheses, group like terms, and add coefficients. To subtract, distribute the minus sign to every term of the second polynomial, then combine like terms.
A sum or difference of polynomials is again a polynomial (closure). Its degree is the larger of the two degrees, unless equal leading terms cancel and drop the degree, as in $(x^2 + x) - (x^2 - 1) = x + 1$ .

Adding and Subtracting Polynomials

Learning goals

What a polynomial is

Standard form

Why only like terms combine

Adding polynomials

Subtracting polynomials

Closure and the degree of a sum

Common mistakes

Practice

Multiple Choice Questions (MCQ)

Free Response Questions (FRQ)