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Factoring Quadratic Polynomials Help (page 2)

By — McGraw-Hill Professional
Updated on Sep 26, 2011

Factoring Shortcuts

Shortcut 1: When x2 is the First Term

There is a factoring shortcut when the first term is x2. 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 x2 – 7x + 6 need to sum to 7: x2 – 7x + 6 = (x – 1)(x – 6). The factors of 6 we need for x2 + 5x + 6 need to sum to 5: x2 + 5x + 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

x2 + 3x – 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: x2 + 3x – 10 = (x + 5)(x – 2).

x2 – 5x – 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).

x2 + 11x + 24: 3 · 8 = 24 and 3 + 8 = 11

x2 + 11x + 24 = (x + 3)(x + 8)

x2 – 9x + 18: 3 · 6 = 18 and 3 + 6 = 9

x2 – 9x + 18 = (x – 3)(x – 6)

x2 + 9x – 36: 3 · 12 = 36 and 12 – 3 = 9

x2 + 9x – 36 = (x + 12)(x – 3)

x2 – 2x – 8: 2 · 4 = 8 and 4 – 2 = 2

x2 – 2x – 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.”

x2 + x + 1          x2 + 14x + 19          x2 – 5x + 10

Quadratic polynomials of the form x2c2 are called the difference of two squares . We can use the shortcut on x2c2 = x2 + 0 xc2 . The factors of c2 must have a difference of 0. This can only happen if they are the same, so the factors of c2 we want are c and c .

Examples

x2 – 9 = (x – 3)(x + 3)        x2 – 100 = (x – 10)(x + 10)

x2 – 49 = (x – 7)(x + 7)     16 – x2 = (4 – x)(4 + x)

When the sign between x2 and c2 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  xncn where n is any even number. The factorization is xncn = (xn /2cn /2)( xn /2 + cn /2). [When n is odd, xncn can be factored also but this factorization will not be covered here.]

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