Polynomials and factoring
Polynomials are the most common expressions in algebra. This page shows how to add, subtract, and multiply them. Then it shows the reverse of multiplying: factoring. Factoring is one of the most useful skills in this course. You will use it to solve equations and to simplify fractions.
What is a polynomial?
A polynomial in ${x}$ is a sum of terms of the form ${ax^k}$, where ${a}$ is a real number and ${k}$ is a whole number (${0,1,2,\ldots}$). For example,
${4x^3-5x^2+x-7}$
is a polynomial with four terms: ${4x^3}$, ${-5x^2}$, ${x}$, and ${-7}$. Here are the words we use:
- The number in front of a term is its coefficient. The coefficient of ${-5x^2}$ is ${-5}$.
- A term with no variable, like ${-7}$, is the constant term.
- The degree of a polynomial is the highest power of ${x}$. The polynomial above has degree ${3}$.
- The coefficient of the highest power is the leading coefficient. Above, it is ${4}$.
- A polynomial with one term is a monomial, with two terms a binomial, and with three terms a trinomial.
We usually write a polynomial in standard form: the powers go down from left to right.
Not every expression is a polynomial. For example, ${x^{-2}+1}$ and ${\sqrt{x}+3}$ are not, because the powers ${-2}$ and ${\tfrac{1}{2}}$ are not whole numbers.
Example 1: Write ${7-2x^4+3x}$ in standard form. Then give its degree and leading coefficient.
Solution:
Put the powers in order, highest first:
${-2x^4+3x+7}$
The highest power is ${x^4}$, so the degree is ${4}$. The leading coefficient is ${-2}$.
Adding and subtracting polynomials
Like terms have the same variable raised to the same power, such as ${3x^2}$ and ${-5x^2}$. To add polynomials, combine like terms by adding their coefficients.
To subtract a polynomial, change the sign of every one of its terms, and then add.
Example 2: Find ${(5x^3-2x^2+4)-(3x^3+x^2-6x+1)}$.
Solution:
The minus sign changes the sign of every term in the second polynomial:
${5x^3-2x^2+4-3x^3-x^2+6x-1}$
Now combine like terms: ${5x^3-3x^3=2x^3}$, ${-2x^2-x^2=-3x^2}$, and ${4-1=3}$. The term ${6x}$ has no like term. So the answer is
${2x^3-3x^2+6x+3}$
Multiplying polynomials
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 again and again.
For two binomials, the four products are often remembered as FOIL: First, Outer, Inner, Last.
Example 3: Multiply.
(a) ${(2x+3)(x-5)}$
(b) ${(x+4)(x^2-2x+1)}$
Solution:
(a) Use FOIL:
- First: ${2x\cdot x=2x^2}$
- Outer: ${2x\cdot(-5)=-10x}$
- Inner: ${3\cdot x=3x}$
- Last: ${3\cdot(-5)=-15}$
Add them, and combine the like terms ${-10x}$ and ${3x}$:
${(2x+3)(x-5)=2x^2-7x-15}$
(b) Multiply each term of ${x^2-2x+1}$ by ${x}$, and then by ${4}$:
$\begin{align*}&(x+4)(x^2-2x+1)\\&=x^3-2x^2+x+4x^2-8x+4\\&=x^3+2x^2-7x+4\end{align*}$
Special products
Some products come up so often that it is worth knowing them by heart:
| Product | Result |
|---|---|
| ${(a+b)(a-b)}$ | ${a^2-b^2}$ |
| ${(a+b)^2}$ | ${a^2+2ab+b^2}$ |
| ${(a-b)^2}$ | ${a^2-2ab+b^2}$ |
| ${(a+b)^3}$ | ${a^3+3a^2b+3ab^2+b^3}$ |
| ${(a-b)^3}$ | ${a^3-3a^2b+3ab^2-b^3}$ |
A common mistake is to write ${(a+b)^2=a^2+b^2}$. The middle term ${2ab}$ is missing. For example, ${(3+4)^2=49}$, but ${3^2+4^2=25}$.
Example 4: Use the special products to multiply.
(a) ${(3x-2)^2}$
(b) ${(x^2+5)(x^2-5)}$
Solution:
(a) Use ${(a-b)^2=a^2-2ab+b^2}$ with ${a=3x}$ and ${b=2}$:
$\begin{align*}&(3x-2)^2\\&=(3x)^2-2(3x)(2)+2^2\\&=9x^2-12x+4\end{align*}$
(b) Use ${(a+b)(a-b)=a^2-b^2}$ with ${a=x^2}$ and ${b=5}$:
${(x^2+5)(x^2-5)=x^4-25}$
Factoring
To factor a polynomial means to write it as a product of simpler polynomials. It is multiplication in reverse. For example, from Example 3,
${2x^2-7x-15=(2x+3)(x-5)}$.
You can always check a factoring by multiplying the factors back out.
Common factors
Always look first for the greatest common factor (GCF): the largest expression that divides every term. Then use the distributive property in reverse to take it out.
Example 5: Factor.
(a) ${6x^3-9x^2}$
(b) ${x^3+2x^2-3x-6}$
Solution:
(a) Each term has a factor ${3}$ and a factor ${x^2}$. So the GCF is ${3x^2}$:
${6x^3-9x^2=3x^2(2x-3)}$
(b) There is no factor common to all four terms. So use grouping: factor the first two terms and the last two terms separately.
$\begin{align*}&x^3+2x^2-3x-6\\&=x^2(x+2)-3(x+2)\end{align*}$
Now ${x+2}$ is a common factor of both groups. Take it out:
$\begin{align*}&=(x+2)(x^2-3)\end{align*}$
Factoring trinomials
To factor ${x^2+bx+c}$, look for two numbers that multiply to ${c}$ and add to ${b}$. If the numbers are ${p}$ and ${q}$, then
${x^2+bx+c=(x+p)(x+q)}$.
When the leading coefficient is not ${1}$, as in ${ax^2+bx+c}$, use the ${ac}$ method:
- Find two numbers that multiply to ${ac}$ and add to ${b}$.
- Use them to split the middle term ${bx}$ into two terms.
- Factor by grouping.
Example 6: Factor.
(a) ${x^2-5x+6}$
(b) ${6x^2+x-2}$
Solution:
(a) We need two numbers that multiply to ${6}$ and add to ${-5}$. They are ${-2}$ and ${-3}$:
${x^2-5x+6=(x-2)(x-3)}$
Check: Multiply the factors back out:
$\begin{align*}&(x-2)(x-3)\\&=x^2-3x-2x+6\\&=x^2-5x+6\end{align*}$
(b) Here ${a=6}$, ${b=1}$, ${c=-2}$. So ${ac=-12}$. We need two numbers that multiply to ${-12}$ and add to ${1}$. They are ${4}$ and ${-3}$.
Split the middle term ${x}$ into ${4x-3x}$. Then group:
$\begin{align*}&6x^2+x-2\\&=6x^2+4x-3x-2\\&=2x(3x+2)-1(3x+2)\\&=(3x+2)(2x-1)\end{align*}$
Special patterns
Read the special products backward, and you get these factoring patterns:
| Pattern | Factored form |
|---|---|
| Difference of squares | ${a^2-b^2=(a+b)(a-b)}$ |
| Perfect square trinomial | ${a^2+2ab+b^2=(a+b)^2}$ ${a^2-2ab+b^2=(a-b)^2}$ |
| Difference of cubes | ${a^3-b^3}$ ${=(a-b)(a^2+ab+b^2)}$ |
| Sum of cubes | ${a^3+b^3}$ ${=(a+b)(a^2-ab+b^2)}$ |
A sum of squares, like ${x^2+4}$, cannot be factored using real numbers.
Example 7: Factor completely.
(a) ${9x^2-25}$
(b) ${4x^2+12x+9}$
(c) ${x^3-8}$
(d) ${x^4-16}$
Solution:
(a) This is a difference of squares, with ${a=3x}$ and ${b=5}$:
${9x^2-25=(3x+5)(3x-5)}$
(b) The first and last terms are squares: ${(2x)^2}$ and ${3^2}$. The middle term is ${2(2x)(3)=12x}$. So this is a perfect square:
${4x^2+12x+9=(2x+3)^2}$
(c) This is a difference of cubes, with ${a=x}$ and ${b=2}$:
${x^3-8=(x-2)(x^2+2x+4)}$
(d) This is a difference of squares, with ${a=x^2}$ and ${b=4}$:
${x^4-16=(x^2+4)(x^2-4)}$
But ${x^2-4}$ is another difference of squares. Factor it too:
${x^4-16=(x^2+4)(x+2)(x-2)}$
The factor ${x^2+4}$ is a sum of squares, so we stop. "Factor completely" means keep going until no factor can be factored again.
A factoring strategy
When you meet a polynomial to factor, work through these steps in order:
- Take out the greatest common factor, if there is one.
- Count the terms.
- Two terms: look for a difference of squares, or a sum or difference of cubes.
- Three terms: look for a perfect square. If not, find two numbers (or use the ${ac}$ method).
- Four terms: try grouping.
- Check each factor. If one can be factored again, factor it.
- Check your answer by multiplying.
Summary
- The degree of a polynomial is its highest power. The leading coefficient is the coefficient of that power.
- Add or subtract polynomials by combining like terms. Multiply by multiplying every term by every term.
- ${(a+b)^2=a^2+2ab+b^2}$, not ${a^2+b^2}$.
- Factoring is multiplying in reverse. Always take out the GCF first.
- Factor trinomials by finding two numbers, or with the ${ac}$ method. Factor four terms by grouping.
- Know the patterns: difference of squares, perfect squares, and sums and differences of cubes. A sum of squares does not factor.