Trinomial Factoring

In trinomial factoring, we are looking for two binomials that multiply to get the given trinomial. Most trinomials have the form ax^{2} + bx + c.

Basic steps for trinomial factoring:

Example 1: Simple trinomial with GCF

^{- }7x^{3} - 7x^{2} + 84x Step 1. Find the GCF, if the trinomial has one. Each term in the trinomial has the factor -7x so let's factor it out. -7x(x^{2} + x - 12) Now we will factor the trinomial x^{2} + x - 12. Step 2. Find the factors of the coefficient of ax^{2}. The coefficient of x^{2} is 1. This means x^{2} has only the factors x and x. The equation looks like -7x(x ... )(x ... ) Step 3. Find the factors of -12. Factors of -12: 1, 2, 3, 4, 6, 12, -1, -2, -3, -4, -6, and -12. Since c is negative, the factors in the last terms have opposite signs. So the equation looks like -7x(x + ... )(x - ... ) Step 4. Guess and check combinations of factors. One way to do this is to put the factors of the coefficients into two columns such that the first column is for a and the second column is for c. For this example, a = 1 and c = -12. In this case, we have the following combinations: 11 -112 11 1-12 11 -26 11 2-6 11 -34 11 3-4 Try each combination by multiplying across. 11 -112 ^{-1} _{12} 11 1-12 ^{1} _{-12} 11 -26 ^{-2} _{6} 11 2-6 ^{2} _{-6} 11 -34 ^{-3} _{4} 11 3-4 ^{3} _{-4} Add the products. Which combination equals b? -1 + 12 = 11 1 + (-12) = -11 -2 + 6 = 4 2 + (-6) = -4 -3 + 4 = 1 3 + (-4) = -1 For the trinomial x^{2} + x - 12, b = 1. 11 -34 is the combination that adds up to 1. This means that x^{2} + x - 12 = (x -3)(x + 4) The answer is .

Step 1. Find the GCF, if the trinomial has one. Each term in the trinomial has the factor -7x so let's factor it out.

-7x(x^{2} + x - 12)

Step 2. Find the factors of the coefficient of ax^{2}. The coefficient of x^{2} is 1.

This means x^{2} has only the factors x and x. The equation looks like -7x(x ... )(x ... )

Since c is negative, the factors in the last terms have opposite signs. So the equation looks like -7x(x + ... )(x - ... )

Example 2: Trinomial where a ≠ 1

4x^{2} - 16x + 15 Step 1. Find the GCF, if the trinomial has one. There is no GCF for this trinomial so we cannot factor anything out. Step 2. Find the factors of the coefficient of ax^{2}. The coefficient of 4x^{2} is 4. Factors of 4: 1, 2, and 4. The equation looks like either (x ... )(4x ... ) or (2x ... )(2x ... ) Step 3. Find the factors of 15. Since c is positive and b is negative, the factors in the last terms have - signs. So the equation looks like either (x - ... )(4x - ... ) or (2x - ... )(2x - ... ) So we will look at the negative factors of 15: -1, -3, -5, and -15. Step 4. Guess and check combinations of factors. One way to do this is to put the factors of the coefficients into two columns such that the first column is for a and the second column is for c. For this example, a = 4 and c = 15. In this case, we have the following combinations: 14 -1-15 14 -3-5 22 -1-15 22 -3-5 Try each combination by multiplying across. 14 -1-15 ^{-4} _{-15} 14 -3-5 ^{-12} _{-5} 22 -1-15 ^{-2} _{-30} 22 -3-5 ^{-6} _{-10} Add the products. Which combination equals b? -4 + (-15) = -19 -12 + (-5) = -17 -2 + (-30) = -32 -6 + (-10) = -16 For the trinomial 4x^{2} - 16x + 15, b = -16. 22 -3-5 is the combination that adds up to -16. This means that 4x^{2} - 16x + 15 = (2x - 3)(2x - 5) The answer is .

Step 1. Find the GCF, if the trinomial has one. There is no GCF for this trinomial so we cannot factor anything out.

Step 2. Find the factors of the coefficient of ax^{2}. The coefficient of 4x^{2} is 4. Factors of 4: 1, 2, and 4.

The equation looks like either (x ... )(4x ... ) or (2x ... )(2x ... )

Step 3. Find the factors of 15.

Since c is positive and b is negative, the factors in the last terms have - signs. So the equation looks like either (x - ... )(4x - ... ) or (2x - ... )(2x - ... ) So we will look at the negative factors of 15: -1, -3, -5, and -15.

Math Score homepage