Math 101 Intermediate Algebra
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Factoring Trinomials
Chapter 6, Section 2

First, always factor out the greatest common factor of all the terms.

What is Factoring a Quadratic Trinomial?

A quadratic trinomial (polynomial of degree 2 in one variable) has the form

ax2 + bx + c, a not 0.

The quadratic trinomial is given.

The goal is to find numbers A, B, C, and D so that

(Ax + B)(Cx + D) = ax2 + bx + c.

Then (Ax + B)(Cx + D) is the factored form of ax2 + bx + c.

Then (Ax + B) and (Cx + D) are the factors of ax2 + bx + c.

How Are the Factors (Ax + B) and (Cx + D) Found?

By trial and error, find numbers A, B, C, and D so that

A · C = a,
A · D + B · C = b, and
B · D = c

This means that
  • A and C must be factors of the coefficient a,
  • B and D must be factors of the coefficient c, and
  • A · D + B · C must equal the coefficient b.

Why?

Multiplying out (Ax + B)(Cx + D) gives that

ACx2 + (AD + BC)x + BD =
ax2 + bx + c.

Equating coefficients on left and right hand sides gives

A · C = a,
A · D + B · C = b, and
B · D = c

Steps to finding A, B, C, and D

  1. Choose A and C such that A · C = a.
  2. Choose B and D such that B · D = c.
  3. Repeat steps 1 and 2 until A · D + B · C = b.

Not all polynomials can be factored. If you have exhausted all the possiblities for A, B, C, and D and been unable to factor the polynomial, then the polynomial cannot be factored.

Factoring General Trinomials

A general trinomial will have more than one variable. For example,

3x2 + 2xy - 8y2

which factors into

(x + 2y)(3x - 4y).

The goal when factoring general trinomials is to find expressions A, B, C, and D such that

(A + B)(C + D) equals the given polynomial.

Note that A, B, C, and D may be expressions containing variables.

Steps to finding A, B, C, and D

  1. Choose A and C such that A · C = the first term.
  2. Choose B and D such that B · D = last term.
  3. Repeat steps 1 and 2 until A · D + B · C = the middle term.

Polynomials with Quadratic Form

Sometimes, higher degree polynomials can be factored when observed to have quadratic form.

A polynomial has quadratic form if the polynomial resulting from a variable substitution is quadratic.

For example, y4 + 5y2 + 6 has quadratic form because the resulting polynomial is quadratic when x is substituted for y2:

  y4 + 5y2 + 6  
= (y2)2 + 5(y2) + 6  
= (x)2 + 5(x) + 6   x is substituted for y2
= x2 + 5x + 6   resulting polynomial

Then x2 + 5x + 6 factors into (x + 2)(x + 3).

Finally, resubstituting y2 back in for x gives

y4 + 5y2 + 6 = (y2 + 2)(y2 + 3)

Remember to check the two new factors to see if they can each be factored further.

If One Factor of a Polynomial is Known....

Use polynomial division (either synthetic or long) to find another factor (i.e., the quotient) of the polynomial.

Try to factor the new factor (i.e., the quotient).