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
Take the expression
Every piece of it follows the same pattern, a number multiplied by a whole-number power of . Reading it piece by piece pins down the vocabulary you will use for the rest of the chapter.
| Term | Coefficient | Degree |
|---|---|---|
The degree of a term is the power of in it. The term is really , so its degree is , and the constant is really (since ), so its degree is . The degree of the polynomial is the largest degree among its terms, which is here. The highest-degree term, , is the leading term, and its coefficient is the leading coefficient. The constant term is the term with no variable, the term, which is . Notice that this polynomial has no term. That simply means its coefficient is , and a term with a zero coefficient contributes nothing and is left out.
In general, a polynomial in is any finite sum of terms built the same way, each one a real number (its coefficient) times a whole-number power of . Every exponent must be a nonnegative integer (). 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.
Check your understanding
Which of these is a polynomial in ?
A polynomial is a finite sum of terms, each a number times a nonnegative-integer power of , exponents only, never negative, never fractional, and never a variable stuck under a root or in a denominator.
fits every part of that test: its exponents are , , and , all nonnegative integers. The other three fail it. has in a denominator, which is really , a negative exponent. has under a root, which is really , a fractional exponent. And writes a negative exponent directly. None of the three qualifies as a polynomial.
Short polynomials get their own names by their number of terms. A one-term polynomial such as or or is a monomial. A two-term polynomial such as is a binomial. A three-term polynomial such as 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 above is already in standard form. You could rearrange it as , and it would equal the same number for every value of , 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 , what is the leading coefficient?
The leading coefficient is the coefficient of the highest-degree term, not the first coefficient written down. The degrees of the terms are , , , and , so the leading term is the term.
Rewriting in standard form as makes the leading term obvious. The is only the coefficient of .
Check your understanding
Which of these is written in standard form?
Standard form lists the terms in descending order of degree: the degree- term first, then degree , then degree , then the constant.
The original order goes constant, degree , degree , degree , which is not sorted by degree at all. And flipping the sign on changes the polynomial itself, not just the order its terms are written in.
Why only like terms combine
Like terms are terms with the same variable part, meaning the same power of . In a single variable that just means the same exponent. So and are like terms, and and are like terms, but and are not, and neither are and . 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, . Let the shared quantity be a single power of , say . Two terms that both carry that power are copies and copies of the very same thing, so
The common factor pulls out to the right, the coefficients and 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 , so the answer is , never .
Now try two different powers, and . Factoring the smaller power out is the most you can do,
and is not a constant, since it still contains an , so the result is not a single term for any constant and power . The two terms cannot merge into one, because they are not copies of the same shared power. 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
The leading plus sign lets the parentheses drop with every sign kept as it stands:
Group the like terms by power, then add the coefficients in each group:
The terms give , the terms give , and the constants give . The sum 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 playing the role that place values play for whole numbers.
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 .
Worked example 2 Add
The first polynomial has no term and the second has no term, so those powers each appear in only one of the two. Drop the parentheses and group by power:
Only the terms and the constants have partners to combine:
The and the carry straight down, because nothing is there to add to them. Writing the result in descending order gives .
Check your understanding
Add .
Drop the parentheses and group by power. The terms combine, the term has no partner and carries straight down, the term has no partner either, and the constants combine.
Adding and gives , not : read the coefficient's sign, do not just add its size. And the lone term still has to carry down; it is easy to drop a term that has no partner.
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 multiplying the whole group. By the distributive property, that factor flips the sign of each term inside:
The becomes , the becomes , and the becomes . Every sign turns over. Once the parentheses are gone, you combine like terms exactly as with addition.
Worked example 3 Subtract
Distribute the minus sign across all three terms of the second polynomial, flipping each sign:
Watch the two signs that changed: became , and became . Now group and combine:
Had you flipped only the first term and written , the term and the constant would both come out wrong. The sign change has to reach every term.
Worked example 4 Subtract
The second polynomial has four terms, so four signs flip. In particular the becomes , and the becomes :
Group by power and add the coefficients in each group:
The middle group is the one to double-check: plus the flipped gives , not . Subtracting a negative term adds it back, which is precisely what the distributed minus sign takes care of.
Check your understanding
Simplify .
Distribute the minus sign to every term of the second polynomial, not just the first. Each of , , and flips its sign.
Stopping after the first flip gives , the most common error. The and the have to flip too, which turns the term into and the constant into .
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. In the rare case where every term cancels, the result is itself, which has no degree at all, since there is no term left to point to.
Worked example 5 Compare the degree of with
Both polynomials have degree , with the same leading term . Add them first:
The leading coefficients are and , and is not zero, so the term survives and the sum keeps degree . Now subtract instead:
This time the terms cancel, since , and the highest power left is . The difference has degree , lower than either polynomial you started with. The same two polynomials give a degree- sum and a degree- 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 ?
Both polynomials have degree , but their leading coefficients, and , are opposites, so adding cancels the term.
With the terms gone, the highest power left is , so the degree is , lower than either original polynomial. Notice this cancellation happened in a sum: the leading coefficients were opposites ( and ), not equal, the way they were in Worked Example 5's subtraction.