Factoring review

There is an analogy between polynomials and the way we write integers in base 10. For example, the integer 253 is like the polynomial 2x² + 5x + 3 (especially if you evaluate it when x = 10). When you add integers, you can arrange the digits in columns and add the columns; when you add polynomials, you can arrange the terms in columns and add the columns. When you multiply integers, you need to multiply each digit in one integer by each digit in the other; when you multiply polynomials, you need to multiply each term in one polynomial by each term in the other. (In some ways, arithmetic with polynomials is easier to understand, since there is no carrying.)

Factoring is running multiplication backwards. To factor the integer 253, you consider how you could get it by multiplying smaller integers; the answer is that 253 = 11 × 23. Similarly, to factor the polynomial 2x² + 5x + 3, you consider how you could get it by multiplying simpler polynomials; the answer is that 2x² + 5x + 3 = (x + 1) ⋅ (2x + 3). (This is not supposed to be obvious, but you can check that it's true by multiplying.) As the integers 11 and 23 are the factors of 253, so the polynomials x + 1 and 2x + 3 are the factors of 2x² + 5x + 3.

Common factors

The simplest technique for factoring is when one of the factors is a constant. When you do this, here are some rules to follow, which apply whenever the original polynomial has only rational coefficients (otherwise it gets more complicated):

There can only be one constant factor, which comes first.
In each non-constant factor, the leading coefficient should be positive.
In each non-constant factor, all of the coefficients should be integers.
In each non-constant factor, the coefficients should not have any common integer factor greater than 1.

Here are some ways to factor 6x − 12 that violate these rules and so are wrong:

6x − 12 = 2(x − 2)3;
6x − 12 = −6(−x + 2);
6x − 12 = 12(½x − 1);
6x − 12 = 2(3x − 6).

All of these equations are identities (they're always true), but they are not the proper way to factor 6x − 12. The only proper way, following all of the rules above, is 6x − 12 = 6(x − 2).

Here are some more examples:

Look for a factor common to each term: 2y + 20 = 2 ⋅ y + 2 ⋅ 10 = 2(y + 10).
Sometimes it helps to factor the coefficients: 6t + 15 = 2 ⋅ 3 ⋅ t + 3 ⋅ 5 = 3(2t + 5).
You can also try to factor out a variable that appears in every term: 5x³ + 3x² = 5xxx + 3xx = x²(5x + 3).
A negative coefficient on the leading term means that a minus sign appears out front: 8 − 2x = −2x + 8 = −2x − −8 = −2(x − 4).
If the coefficients are fractional, then their common denominator appears in the denominator out front: ½x + ⅔ = 3⁄6 ⋅ x + 4⁄6 = ⅙(3x + 4).

Sometimes the textbook is a little sloppy about fractional and negative coefficients, and it's true that the rules are somewhat arbitrary; but you need to pick some rules and follow them consistently in order to guarantee that everything will work out in a complicated problem.

Summary of factoring techniques

Here are the steps for factoring polynomials in Beginning and Intermediate Algebra:

If necessary, put the polynomial in standard form.
If possible, pull out any factors common to all terms (§6.1).
If there are four terms, try factoring by grouping (§6.1).
If there are three terms, try factoring into two binomials (§§6.2&6.3) or factoring as a perfect square (§6.4).
If there are two terms (or if you now have factors with two terms), try factoring as a sum or difference of squares or cubes (§6.4).
Keep factoring the factors until you can factor no further (§6.5).

These techniques will work for all polynomials up to degree 2 and for some polynomials of higher degree.

For definiteness, here are the conditions that must be met for a polynomial (with rational coefficients) to be completely factored:

The first factor must be a constant, except that (unless it is the only factor) we leave it out if it is 1 or use just a minus sign if it is −1.
Every other factor must be a non-constant polynomial with integer coefficients and a positive leading coefficient.
No factor's coefficients may have a common integer factor greater than 1.
No factor may be a product of two non-constant polynomials.

The last of these is the one that can be hard to check and may require fancy techniques to fix.

A product of two non-constant polynomials is called a composite polynomial. (The last rule above requires us to factor these polynomials further.) A non-constant polynomial that is not composite is called a prime polynomial. (The constant polynomials are considered neither prime nor composite.) Compare that a product of two whole numbers greater than 1 is called a composite number, while a whole number greater than 1 that is not composite is called a prime number. (The whole numbers 0 and 1 are neither prime nor composite. In this analogy, the non-zero constant polynomials correspond to the whole number 1, while the constant polynomial 0 corresponds to the whole number 0.)

Solving equations by factoring

Here are the steps for solving equations by factoring:

Subtract the right-hand side of the equation away, moving it to the left-hand side, so that the right-hand side of the equation is zero.
Simplify the left-hand side of the equation and factor it.
Split the equation up into several equations (with ‘or’ between them), one equation for each factor, each factor set equal to zero.
Solve these equations; throw out any which have no solution.
If any solutions are repeated, count them only once.
List all possible solutions (as a list of statements with ‘or’ between them or in a solution set if you prefer).

If you can figure out how to do the factoring step, then this method will solve any polynomial equation in one variable, as long as all of the solutions are rational numbers; and it may help at other times as well.

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