**Factoring Shortcuts**

**Shortcut 1: When **** ***x*^{2 }is the First Term

**is the First Term**

*x*^{2 }There is a factoring shortcut when the first term is *x*^{2}. If the second sign is plus, choose the factors whose *sum* is the coefficient of the second term. For example the factors of 6 we need for *x*^{2} – 7*x* + 6 need to sum to 7: x^{2} – 7*x* + 6 = (*x* – 1)(*x* – 6). The factors of 6 we need for *x*^{2} + 5*x* + 6 need to sum to 5: *x*^{2} + 5*x* + 6 = (*x* + 2)(*x* + 3).

If the second sign is minus, the *difference* of the factors needs to be the coefficient of the middle term. If the first sign is plus, the bigger factor will have the plus sign. If the first sign is minus, the bigger factor will have the minus sign.

**Examples**

x^{2} + 3*x* – 10: The factors of 10 whose difference is 3 are 2 and 5. The first sign is plus, so the plus sign goes with 5, the bigger factor: *x*^{2} + 3*x* – 10 = (*x* + 5)(*x* – 2).

x^{2} – 5*x* – 14: The factors of 14 whose difference is 5 are 2 and 7. The first sign is minus, so the minus sign goes with 7, the bigger factor: *x* ^{2} – 5 *x* – 14 = (*x* – 7)(*x* + 2).

x^{2} + 11*x* + 24: 3 · 8 = 24 and 3 + 8 = 11

x^{2} + 11*x* + 24 = (*x* + 3)(*x* + 8)

x^{2} – 9*x* + 18: 3 · 6 = 18 and 3 + 6 = 9

x^{2} – 9*x* + 18 = (*x* – 3)(*x* – 6)

x^{2} + 9*x* – 36: 3 · 12 = 36 and 12 – 3 = 9

x^{2} + 9*x* – 36 = (*x* + 12)(*x* – 3)

x^{2} – 2*x* – 8: 2 · 4 = 8 and 4 – 2 = 2

x^{2} – 2*x* – 8 = (*x* + 2)(*x* – 4)

Find practice problems and solutions at Factoring Quadratic Polynomials Practice Problems - Set 3.

**Shortcut 2: The Difference of Two Squares**

This shortcut can help you identify quadratic polynomials that do not factor “nicely” without spending too much time on them. The next three examples are quadratic polynomials that do not factor “nicely.”

x^{2} + *x* + 1 x^{2} + 14*x* + 19 x^{2} – 5*x* + 10

Quadratic polynomials of the form *x*^{2} – *c*^{2} are called the *difference of two squares* . We can use the shortcut on *x*^{2} – *c*^{2} = *x*^{2} + 0 *x* – *c*^{2} . The factors of *c*^{2} must have a difference of 0. This can only happen if they are the same, so the factors of *c*^{2} we want are *c* and *c* .

**Examples**

x^{2} – 9 = (*x* – 3)(*x* + 3) x^{2} – 100 = (*x* – 10)(*x* + 10)

x^{2} – 49 = (*x* – 7)(*x* + 7) 16 – *x*^{2} = (4 – *x*)(4 + *x*)

When the sign between *x*^{2} and *c*^{2} is plus, the quadratic cannot be factored using real numbers.

Find practice problems and solutions at Factoring Quadratic Polynomials Practice Problems - Set 4.

**Shortcut 3: The Difference of Two Squares with Even Coefficients**

The difference of two squares can come in the form *x*^{n} – *c*^{n} where *n* is any even number. The factorization is *x*^{n} – *c*^{n} = (*x*^{n /2} – *c*^{n /2})( *x*^{n /2} + *c*^{n /2}). [When n is odd, *x*^{n} – *c*^{n} can be factored also but this factorization will not be covered here.]

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