Polynomials

Factoring Trinomials

Factoring Trinomials of the Form x ² + bx + c and x ² - bx + c

Just as the product of two binomials can often be rewritten as a trinomial, trinomials of the form ax ² + bx + c can often be rewritten as the product of two binomials. For example, x ² + 3x + 2 = (x + 1)(x + 2) .

We now know that the product of two binomials of the form (x + d ) and (x + e) is given by:

Thus, in order to rewrite a binomial x ² + bx + c as the product of two binomials ( b positive or negative), we must find numbers d and e such that d + e = b and de = c . Since c is positive, d and e must have the same sign.

Here are the steps to factoring a trinomial of the form x ² + bx + c , with c > 0 . We assume that the coefficients are integers, and that we want to factor into binomials with integer coefficients.

1. Write out all the pairs of numbers which can be multiplied to produce c .
2. Add each pair of numbers to find a pair that produce b when added. Call the numbers in this pair d and e .
3. If b > 0 , then the factored form of the trinomial is (x + d )(x + e) . If b < 0 , then the factored form of the trinomial is (x - d )(x - e) .
4. Check: The binomials, when multiplied, should equal the original trinomial.

Note: Some trinomials cannot be factored. If none of the pairs total b , then the trinomial cannot be factored.

Example 1: Factor x ² + 5x + 6 .

1. Pairs of numbers which make 6 when multiplied: (1, 6) and (2, 3) .
2. 1 + 6≠5 . 2 + 3 = 5. Thus, d = 2 and e = 3 (or vice versa).
3. (x + 2)(x + 3)
4. Check: (x + 2)(x + 3) = x ² +3x + 2x + 6 = x ² + 5x + 6

Thus, x ² + 5x + 6 = (x + 2)(x + 3) .

Example 2: Factor x ² - 7x + 12 .

1. Pairs of numbers which make 12 when multiplied: (1, 12) , (2, 6) , and (3, 4) .
2. 1 + 12≠7 . 2 + 6≠7 . 3 + 4 = 7. Thus, d = 3 and e = 4 .
3. (x - 3)(x - 4)
4. Check: (x - 3)(x - 4) = x ² -4x - 3x + 12 = x ² - 7x + 12

Thus, x ² - 7x + 12 = (x - 3)(x - 4) .

Example 3: Factor 2x ³ +4x ² + 2x .
First, remove common factors: 2x ³ +4x ² +2x = 2x(x ² + 2x + 1)

1. Pairs of numbers which make 1 when multiplied: (1, 1) .
2. 1 + 1 = 2. Thus, d = 1 and e = 1 .
3. 2x(x + 1)(x + 1) (don't forget the common factor!)
4. Check: 2x(x + 1)(x + 1) = 2x(x ² +2x + 1) = 2x ³ +4x ² + 2x

Thus, 2x ³ +4x ² +2x = 2x(x + 1)(x + 1) = 2x(x + 1)² .
x ² + 2x + 1 is a perfect square trinomial.

Factoring Trinomials of the Form x ² + bx - c and x ² - bx - c

In the equation (x + d )(x + e) = x ² + (d + e)x + de = x ² + bx + c (mentioned in Heading ), c is negative if and only if de is negative; that is, if and only if d is negative or e is negative. Thus, the equation becomes (x + d )(x - e) = x ² + (d - e)x - de = x ² + bx - c , which c > 0 .

In order to rewrite the trinomial x ² + bx - c as the product of two binomials, we must find numbers d and e such that d - e = b and | de| = c . If b is positive, then d - e > 0 , so d > e . If b is negative, then d - e < 0 , so d < e .

Here are the steps to factoring a trinomial of the form x ² + bx - c , with c > 0 :

1. Write out all the pairs of numbers which can be multiplied to produce c .
2. Subtract each pair of numbers to find a pair that produce b when one is subtracted from the other. Call the numbers in this pair d and e . If b > 0 , let d be the larger number, and if b < 0 , let d be the smaller number.
3. The factored form of the trinomial is (x + d )(x - e) .
4. Check: The binomials, when multiplied, should equal the original trinomial. If the middle term has the wrong sign, you most likely switched d and e . Switch the "+" and "-" sign in your binomials and check again.

Note: Again, not every trinomial can be factored.

Example 1: Factor x ² + 6x - 16 .

1. Pairs of numbers which make 16 when multiplied: (1, 16) , (2, 8) , and (4, 4) .
2. 16 - 1≠6 . 8 - 2 = 6. Since b = 6 > 0 , d = 8 and e = 2 .
3. (x + 8)(x - 2)
4. Check: (x + 8)(x - 2) = x ² -2x + 8x - 16 = x ² + 6x - 16

Thus, x ² + 6x - 16 = (x + 8)(x - 2) .

Example 2: Factor x ² - x - 20 .

1. Pairs of numbers which make 20 when multiplied: (1, 20) , (2, 10) , and (4, 5) .
2. 20 - 1≠1 . 10 - 2≠1 . 5 - 4 = 1. Since b = - 1 < 0 , d = 4 and e = 5 .
3. (x + 4)(x - 5)
4. Check: (x + 4)(x - 5) = x ² -5x + 4x - 20 = x ² - x - 20

Thus, x ² - x - 20 = (x + 4)(x - 5) .

Example 3: Factor x ² - 16 (Here, x has an implied coefficient of b = 0 ).

1. Pairs of numbers which make 16 when multiplied: (1, 16) , (2, 8) , and (4, 4) .
2. 16 - 1≠ 0 . 8 - 2≠ 0 . 4 - 4 = 0. Thus, d = 4 and e = 4 .
3. (x + 4)(x - 4)
4. Check: (x + 4)(x - 4) = x ² -4x + 4x - 16 = x ² - 16

Thus, x ² - 16 = (x + 4)(x - 4) .