What Is a Polynomial?
A term is a number, a variable, or a number multiplied by variables raised to whole-number powers. Examples of terms: , , , .
A polynomial is a sum of terms. In each term:
- the coefficient is the number multiplied by the variable (in the coefficient is );
- a constant is a term with no variable (like in ).
A polynomial is a sum of terms, each a coefficient times a power of the variable. Adding and subtracting collect like terms; multiplying uses the distributive property on every pair of terms (FOIL is the two-by-two case). The degree is the highest power and it controls the shape and the number of roots.
Polynomials are exactly the expressions built from a variable using only adding, subtracting and multiplying. That is why the exponents must be whole numbers ( needs division, needs a root), and why polynomials are closed under , and : combining two of them never produces anything else.
- Monomial — one term:
- Binomial — two terms:
- Trinomial — three terms:
- Polynomial — the general word for one or more terms (four or more terms usually just get called “a polynomial”).
This is a trinomial (three terms).
- Terms: , ,
- Coefficients: (of ) and (of )
- Constant:
Calling a polynomial because it "has terms." has a negative exponent, which requires division by the variable. It is a rational expression, not a polynomial.
Watch the sign! The term has a negative coefficient. Always carry the sign in front of a term along with the term.
Which are polynomials: , , , ? Name the polynomials by number of terms.
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is a polynomial (a binomial; fractional coefficients are fine). is not. is a polynomial (a monomial of degree ). is not (negative exponent).
Degree, Leading Coefficient, and Standard Form
The degree of a term is the exponent on its variable ( has degree ; a constant like has degree ).
The degree of a polynomial is the largest degree among its terms.
The degree matters because the highest power eventually outgrows all the others. For at , the cube contributes while the rest contribute . So the leading term alone predicts the shape far from the origin, and standard form puts it first where you can see it.
A polynomial is in standard form when its terms are written in descending order of degree (highest power first).
The leading coefficient is the coefficient of the first term once the polynomial is in standard form (the coefficient of the highest-degree term).
Order the terms from highest power to lowest:
- Degree of the polynomial:
- Leading coefficient:
- Constant:
Reading the leading coefficient from the first term as written: for , "the leading coefficient is ." Leading means highest degree: the term is , so the leading coefficient is .
Standard form: .
- Degree:
- Leading coefficient: (the coefficient of )
Tip: A missing power is fine— simply has no , , or terms. Standard form only cares about ordering the terms that are there.
Write in standard form; give the degree, leading coefficient and constant term.
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Order by degree: . Degree , leading coefficient , constant term .
Evaluating a Polynomial
To evaluate a polynomial for a given value, replace every variable with that value (use parentheses!), then follow the order of operations.
Evaluating a polynomial at a number gives one point on its graph: the input is the -coordinate and the value is the -coordinate. That is why parentheses around a substituted negative matter so much; the sign of versus decides which point you plot.
Evaluating at as . The expression means : square first, , then take the opposite, .
Careful with negatives: means “take the opposite of .” Square first: ; then apply the minus: . Always wrap the substituted value in parentheses.
Evaluate at , and .
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. . . All three are zeros of , points where its graph crosses the -axis.
Adding Polynomials
Like terms have the exact same variable part (same variable, same exponent), such as and . To add polynomials, drop the parentheses and combine like terms.
Adding polynomials is adding like terms, and like terms combine by the distributive property: . Aligning terms by degree, the way digits are aligned by place value, keeps the matching visible.
Group like terms, then add:
Combining unlike powers: . Only terms with the same power combine; is already simplified.
Only like terms combine. You cannot add and into one term—different powers stay separate. The answer is fully simplified.
Add .
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Group by degree: : ; : ; : ; constants: . Sum: .
Subtracting Polynomials
To subtract, distribute the minus sign to every term of the second polynomial (this flips each sign), and then combine like terms.
Subtracting a polynomial is adding its opposite, and the opposite of a sum is the sum of the opposites: . That is the distributive property with the factor , which is why every sign in the second polynomial flips, not only the first.
Distribute the negative across the second group:
Changing only the first sign: as . Every term flips: , giving .
The most common mistake! Subtracting changes all three signs: , , . It is easy to flip only the first term—flip them all.
Subtract .
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Flip every sign of the second: . Combine: .
Multiplying a Monomial by a Polynomial
Multiply the monomial by each term inside the parentheses. Multiply the coefficients, and add the exponents on matching variables ().
The monomial multiplies each term because of the distributive property, and each little product uses the product rule for exponents: . As an area, it is one rectangle of height cut into pieces of widths , and .
Distribute to each term:
Forgetting to multiply the exponents' bases: ("multiply everything"). Coefficients multiply () but exponents add: , so .
Exponent reminder: . Add exponents when multiplying powers of the same base—do not multiply them.
Multiply .
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; ; . Result: .
Multiplying Two Binomials (FOIL)
FOIL names the four products you make when multiplying two binomials:
Then combine the two middle (Outer Inner) like terms.
FOIL is the distributive property used twice: . The box shows why there are four products: each of the two terms in one factor meets each of the two terms in the other. FOIL is only a way to remember all four; for bigger products the box is safer.
Multiplying only first and last terms: . The outer and inner products, and , are missing. The full product is ; check with : and .
FOIL is just the distributive property twice. Every term in the first binomial multiplies every term in the second. That idea is what lets us multiply bigger polynomials next.
Multiply and .
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. .
Multiplying a Binomial by a Trinomial
Multiply each term of the binomial by each term of the trinomial (that is small products), then combine like terms and write in standard form.
Every term of the first factor must meet every term of the second, so a -term times a -term product has pieces before combining, the counting principle in algebra. A box with rows and columns is a checklist that no product gets skipped.
Distribute , then distribute :
Stopping after the first row: . The must multiply all three terms too, adding .
Stay organized. Line up like terms in columns as you go, and count your products—for a binomial times a trinomial you should form exactly six before combining.
Multiply .
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and . Add: the and terms cancel, leaving . (This is the sum-of-cubes pattern.)
Special Products
These come straight from FOIL, but recognizing the pattern saves time.
Both patterns are FOIL with a shortcut built in. In the outer and inner products are and , which cancel, leaving . In they are twice, giving the middle term . The picture shows the first: a square of side with a -square removed can be cut and rearranged into an -by- rectangle.
Here and , so :
Check with FOIL: . The middle terms cancel.
Squaring each term: or . The square of a difference has a middle term: . Test : , and .
Here and , so :
Do NOT drop the middle term! .
The correct expansion is . For example, , not . The term is the one students forget most often.
Expand and using the patterns.
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. With , : .
Going Deeper: Advanced Polynomials
FOIL and repeated distribution get slow for high powers. The Binomial Theorem expands all at once:
where the binomial coefficient counts the ways to choose of the factors that contribute a .
Reading the pattern for a fixed : the power of starts at and drops by one each term, the power of starts at and rises by one, and the two exponents in every term add up to .
The coefficients are exactly the rows of Pascal's triangle. Each entry is the sum of the two entries above it, and every row begins and ends with .
So : the row becomes the coefficients directly.
Use row of Pascal's triangle: . Take , :
The coefficient is not just the Pascal number. In each term also carries the powers of the constants and . The Pascal entry became once and were multiplied in. Always raise both pieces of the binomial to their powers.
Sometimes you only need a single coefficient of a product. Instead of multiplying everything out, collect only the pairs of terms whose degrees add up to the power you want.
To find the coefficient of in a product, list every way to pick one term from each factor so their exponents sum to , multiply those coefficients, and add the results.
We need every pair of terms whose exponents add to :
So the coefficient of is , found without computing any other term.
The sum and product contain enough information to rebuild higher symmetric expressions. Starting from and , rearrange:
The second identity also factors as ; both give the same value.
Substitute directly into the identities:
(Check: give , , and indeed , .)
The perfect-square pattern generalizes to three terms. Every term squares, and every distinct pair produces a doubled cross term:
There are three squares and three cross terms because there are three ways to pair up , , and .
Count the cross terms. A common error is writing only . With terms there are squares plus one doubled cross term for each pair. For example, .
For large , a polynomial behaves like its leading term alone—the highest-degree term dominates every other. So the degree and the sign of the leading coefficient decide the end behavior:
- Even degree: both ends point the same way—up/up if the leading coefficient is positive, down/down if negative.
- Odd degree: the ends point opposite ways—down/up if positive, up/down if negative.
Ignore all but the leading term . The degree is odd () and the leading coefficient is negative (), so the ends point opposite ways: as the value , and as the value (up on the left, down on the right). The large term changes the middle shape but never the ends.
When a polynomial does not divide evenly, long division splits it into a quotient plus a remainder, mirroring long division of numbers. At each step: divide the leading term of the current dividend by the leading term of the divisor, multiply back, and subtract.
So , i.e. the quotient is with remainder .
Keep everything in standard form and hold placeholders. Before dividing, order both polynomials by descending degree and insert coefficients for any missing powers (e.g. write as ) so like terms line up during each subtraction.
Why It Works
Multiplying polynomials is the distributive property applied repeatedly: . The area model makes it geometric — a rectangle with sides and contains four sub-rectangles, one per product. Special products like are pictures of a square cut into a square, two equal rectangles, and a smaller square.
Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.
Formulas, Proofs & Tips
What it means. Multiplying a sum multiplies each piece of it.
Example. .
Why it works. is added times. Regrouping those copies gives added times plus added times, i.e. . It is also the area of an rectangle split into two.
Tip. Distribute the sign too: . Run it backwards to factor.
What it means. The highest-power term decides what the graph does far left and far right.
Example. rises to the left and falls to the right (odd degree, negative lead).
Why it works. For very large , dwarfs every lower power, so the leading term dominates the sum. Multiplying polynomials multiplies their leading terms, adding the exponents.
Tip. Even degree: both ends go the same way. Odd degree: opposite ways. A positive leading coefficient sends the right end up.