⚔️ Algebra Kingdom · Algebra

Polynomials

Name a polynomial by its degree and number of terms, add and subtract polynomials, multiply a monomial or a binomial through, and recognise the special products.

In short

The degree of a polynomial is its largest exponent, and the number of terms gives it its name: monomial, binomial, trinomial.
Standard form writes the terms from the highest power of x down to the constant.
Collecting like terms adds the coefficients and leaves the exponents alone, so $x 2 + x 2$ is $2x 2$ .
A minus sign in front of a bracket changes the sign of every term inside it, not just the first.
Multiplying brackets means every term meets every term, and $(a + b) 2$ always carries the middle term 2ab.

Terms, coefficients, degree, standard form

A polynomial is a sum of terms, and each term is a number multiplied by a whole-number power of a letter: $4x 3, -7x, 9$ . Nothing else is allowed inside one — no letter underneath a division line, no letter under a square root, no negative powers.

Four words describe every polynomial you will meet:

term — a chunk separated by $+$ or -, so $5x 2$ - $3x + 8$ has three of them
coefficient — the number in front of a term, the 5 in $5x 2$
degree — the largest exponent anywhere in the polynomial
standard form — the terms written from the highest power down to the lowest

The degree is about exponents, never about how many terms there are. In 7 - $2x 3 + x$ the degree is 3, even though the $x 3$ is written second and there are three terms.

The number of terms gives the polynomial its name: one term is a monomial, two a binomial, three a trinomial. So $4x 2$ - 9 is a binomial of degree 2.

Standard form matters because it puts the term that matters most — the one that grows fastest — where you can see it first.

Adding and subtracting: collecting like terms

Adding polynomials is nothing new. Drop the brackets, then collect like terms — terms with exactly the same power of x.

$(3x 2 + 5x$ - $1) + (x 2$ - $2x + 6) = 4x 2 + 3x + 5$

The powers never change. Collecting adds the coefficients only: $x 2 + x 2$ is $2x 2$ , in the same way that one crate plus one crate is two crates, not one crate squared.

Subtraction is where marks are lost, because a minus sign in front of a bracket belongs to everything inside it:

$(5x 2 + 2x$ - 3) - $(2x 2$ - $6x + 4)$
$= 5x 2 + 2x$ - 3 - $2x 2 + 6x$ - 4
$= 3x 2 + 8x$ - 7

Notice that -6x became $+6x$ and $+4$ became $-4. A$ safe habit is to rewrite the second bracket with every sign flipped before you collect anything, so the subtraction is finished before the collecting starts.

Multiplying: distribute, or draw the rectangle

To multiply a monomial through a bracket, share it across every term. Multiply the numbers, then the letters — and when two powers of x meet, the exponents add:

$3x 2 (2x 3$ - $5x + 4) = 6x 5$ - $15x 3 + 12x 2$

Multiplying two binomials is the same idea done twice: every term in the first bracket must meet every term in the second, which is four products.

$(x + 3)(2x$ - $1) = 2x 2$ - $x + 6x$ - $3 = 2x 2 + 5x$ - 3

Some people remember this as FOIL — First, Outer, Inner, Last. The area model is the same thing drawn: a rectangle $(x + 3)$ by (2x - 1), cut at each sign into four smaller rectangles whose areas are the four products. The picture makes it obvious why none of the four may be skipped, and why the outer and inner pieces collect into a single middle term.

Three products worth memorising

Three products turn up so often that recognising them saves real time:

$(a + b) 2 = a 2 + 2ab + b 2$
(a - $b) 2 = a 2$ - $2ab + b 2$
$(a +$ b)(a - $b) = a 2$ - $b 2$

The first two are not "square each bit". $(x + 5) 2$ means $(x + 5)(x + 5)$ , and the outer and inner products are both 5x, so they add to a middle term of $10x: x 2 + 10x + 25$ . Writing $x 2 + 25$ is the single most common error in this whole topic.

The third one works the other way round: the outer product is -bx and the inner is $+bx$ , so they cancel and no middle term survives. $(3x + 4)(3x$ - $4) = 9x 2$ - 16.

Note what the sign does. In a square, only the middle term changes sign — the last term is a square, so it stays positive either way.

Where this leads

Every expansion here has a reverse, and that reverse is factoring. Multiplying takes $(x + 2)(x + 5)$ to $x 2 + 7x + 10$ ; factoring starts from $x 2 + 7x + 10$ and finds the brackets again.

That is why the special products are worth knowing by heart in both directions. Meeting $x 2$ - 49 later, you will recognise a difference of two squares and write (x - $7)(x + 7)$ without any searching.

It also explains the one check that never lets you down: multiply your answer back out. If expanding your brackets does not rebuild the polynomial you started with, something in between went wrong.

Polynomials go on to describe areas, volumes, profits and paths through the air. A rectangle whose sides are $(x + 4)$ and $(x + 1)$ has area $x 2 + 5x + 4$ — one expression covering every rectangle of that shape at once, whatever x turns out to be.

Worked examples

Example 1

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

The minus in front of the second bracket applies to all three of its terms, so flip every sign inside it: $-x 2$ - $5x + 2$ .
Rewrite the whole thing with no brackets: $4x 2$ - $3x + 7$ - $x 2$ - $5x + 2$ .
Collect the $x 2$ terms: 4 - $1 = 3$ , giving $3x 2$ .
Collect the x terms: -3 - $5 = -8$ , giving -8x.
Collect the constants: $7 + 2 = 9$ . The answer is $3x 2$ - $8x + 9$ .

Example 2

Expand and simplify $(2x + 5)(3x$ - 4).

First terms: 2x times $3x = 6x 2$ .
Outer terms: 2x times $-4 = -8x$ .
Inner terms: 5 times $3x = 15x$ .
Last terms: 5 times $-4 = -20$ .
Collect the two middle terms: $-8x + 15x = 7x$ , so the answer is $6x 2 + 7x$ - 20.

Example 3

A rectangular garden measures $(x + 6)$ metres by $(x + 2)$ metres. Write a simplified expression for its area, then expand $(x + 6) 2$ to check what a square garden of the longer side would cover.

Area $= (x + 6)(x + 2)$ , so multiply the brackets out: x times $x = x 2$ .
The two middle products are 2x and 6x, which collect to 8x, and the last product is 12. The area is $x 2 + 8x + 12$ square metres.
For the square, $(x + 6) 2$ fits $(a + b) 2 = a 2 + 2ab + b 2$ with $a = x$ and $b = 6$ .
$a 2 = x 2, b 2 = 36$ , and the middle term is 2 times x times $6 = 12x$ .
So a square garden of side $(x + 6)$ covers $x 2 + 12x + 36$ square metres — more than the rectangle, as it should be.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

What is the degree of the polynomial $3x 2 + 6$ ?

Answer: 2

Take the terms one at a time: $3x 2$ has exponent $2, 6$ has exponent 0.
The largest exponent is 2.
So the degree is 2, and in standard form the polynomial is $3x 2 + 6$ .

Problem 2

Difficulty 3 of 5

Simplify: $(-8x 2 + 10x + 6)$ - $(5x 2$ - $9x + 8)$

Answer: $-13x 2 + 19x$ - 2

Distribute the minus across the whole bracket: $-8x 2 + 10x + 6$ - $5x 2 + 9x$ - 8
$x 2$ terms: -8 - $5 = -13$
x terms: $10 + 9 = 19$
Constant terms: 6 - $8 = -2$
Result: $-13x 2 + 19x$ - 2

Problem 3

Difficulty 4 of 5

Expand: $8x 2 (2x 2$ - 3x - 1)

Answer: $16x 4$ - $24x 3$ - $8x 2$

Distribute $8x 2$ across all 3 terms.
$8x 2$ times $2x 2 = 16x 4$
$8x 2$ times $(-3x) = -24x 3$
$8x 2$ times $(-1) = -8x 2$
Putting the pieces together: $16x 4$ - $24x 3$ - $8x 2$

Common mistakes

Forgetting the middle term: writing $(x + 5) 2$ as $x 2 + 25$ instead of $x 2 + 10x + 25$ .
Not distributing the minus: turning (5x - 3) - (2x - 4) into 3x - 7 instead of $3x + 1$ .
Adding the exponents when collecting like terms, so $x 2 + x 2$ becomes $x 4$ rather than $2x 2$ .
Multiplying the coefficients only: reading $3x 2$ times 2x as 6x rather than $6x 3$ .
Leaving out one of the two middle products in a FOIL expansion, so only three of the four products are counted.
Reading the degree off the first term written, so 7 - $2x 3 + x$ looks like degree 0 instead of degree 3.

What you should be able to do

Write a polynomial in standard form and state its degree and its number of terms.
Add and subtract polynomials by combining like terms.
Multiply a monomial by a polynomial and a binomial by a binomial.
Expand the special products $(a + b) 2, (a$ - $b) 2$ and $(a +$ b)(a - b).

Where this fits in the curriculum

Common Core

HSA-APR.A.1
High school — Understand that polynomials form a system closed under addition, subtraction and multiplication; add, subtract and multiply polynomials.
HSA-SSE.A.1.A
High school — Interpret parts of an expression, such as terms, factors and coefficients.
HSA-SSE.A.2
High school — Use the structure of an expression to identify ways to rewrite it, for example seeing x^4 - y^4 as a difference of squares.

Ontario

MTH1W.C1.4
Grade 9 de-streamed — Simplify algebraic expressions by applying properties of operations of numbers, using various representations and tools, in different contexts.
MTH1W simplifies expressions by collecting like terms and multiplying monomials; multiplying two binomials and the special products are Grade 10 (MPM2D), so the tag covers the first half of this skill.

SAT

Passport to Advanced Math
Adding, subtracting and multiplying polynomials, including the special products.