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:
(x + d )(x + e) = x^{2} + xe + dx + de = x^{2} + (d + e)x + de |
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.
- Write out all the pairs of numbers which can be multiplied to produce c.
- Add each pair of numbers to find a pair that produce b when added. Call the numbers in this pair d and e.
- 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).
- Check: The binomials, when multiplied, should equal the original trinomial.
Example 1: Factor x^{2} + 5x + 6.
- Pairs of numbers which make 6 when multiplied: (1, 6) and (2, 3).
- 1 + 6≠5. 2 + 3 = 5. Thus, d = 2 and e = 3 (or vice versa).
- (x + 2)(x + 3)
- Check: (x + 2)(x + 3) = x^{2} +3x + 2x + 6 = x^{2} + 5x + 6
Example 2: Factor x^{2} - 7x + 12.
- Pairs of numbers which make 12 when multiplied: (1, 12), (2, 6), and (3, 4).
- 1 + 12≠7. 2 + 6≠7. 3 + 4 = 7. Thus, d = 3 and e = 4.
- (x - 3)(x - 4)
- Check: (x - 3)(x - 4) = x^{2} -4x - 3x + 12 = x^{2} - 7x + 12
Example 3: Factor 2x^{3} +4x^{2} + 2x.
First, remove common factors: 2x^{3} +4x^{2} +2x = 2x(x^{2} + 2x + 1)
- Pairs of numbers which make 1 when multiplied: (1, 1).
- 1 + 1 = 2. Thus, d = 1 and e = 1.
- 2x(x + 1)(x + 1) (don't forget the common factor!)
- Check: 2x(x + 1)(x + 1) = 2x(x^{2} +2x + 1) = 2x^{3} +4x^{2} + 2x
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 ore 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:
- Write out all the pairs of numbers which can be multiplied to produce c.
- 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.
- The factored form of the trinomial is (x + d )(x - e).
- 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.
Example 1: Factor x^{2} + 6x - 16.
- Pairs of numbers which make 16 when multiplied: (1, 16), (2, 8), and (4, 4).
- 16 - 1≠6. 8 - 2 = 6. Since b = 6 > 0, d = 8 and e = 2.
- (x + 8)(x - 2)
- Check: (x + 8)(x - 2) = x^{2} -2x + 8x - 16 = x^{2} + 6x - 16
Example 2: Factor x^{2} - x - 20.
- Pairs of numbers which make 20 when multiplied: (1, 20), (2, 10), and (4, 5).
- 20 - 1≠1. 10 - 2≠1. 5 - 4 = 1. Since b = - 1 < 0, d = 4 and e = 5.
- (x + 4)(x - 5)
- Check: (x + 4)(x - 5) = x^{2} -5x + 4x - 20 = x^{2} - x - 20
Example 3: Factor x^{2} - 16 (Here, x has an implied coefficient of b = 0).
- Pairs of numbers which make 16 when multiplied: (1, 16), (2, 8), and (4, 4).
- 16 - 1≠ 0. 8 - 2≠ 0. 4 - 4 = 0. Thus, d = 4 and e = 4.
- (x + 4)(x - 4)
- Check: (x + 4)(x - 4) = x^{2} -4x + 4x - 16 = x^{2} - 16