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Factoring Quadratic Polynomials Help (page 3)

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

Examples

Factoring Quadratic Polynomials Examples

Find practice problems and solutions at Factoring Quadratic Polynomials Practice Problems - Set 5.

Shortcut 4: Factor Out the Coefficient

When the first term is not x2, see if you can factor out the coefficient of x2. If you can, then you are left with a quadratic whose first term is x2. For example each term in 2x2 + 16x – 18 is divisible by 2:

2x2 + 16x – 18 = 2(x2 + 8x – 9) = 2(x + 9)(x – 1).

Find practice problems and solutions at Factoring Quadratic Polynomials Practice Problems - Set 6.

A Combination of Factoring Methods

The coefficient of the x2 term will not always factor away. In order to factor quadratics such as 4x2 + 8x + 3 you will need to try all combinations of factors of 4 and of 3: (4x + __)( x + __) and (2x + __)(2x + __). The blanks will be filled in with the factors of 3. You will need to check all of the possibilities: (4x + 1)(x + 3), (4x + 3)(x + 1), and (2x + 1)(2x + 3).

Example

4x2 – 4x – 15

The possibilities to check are

(a) (4x + 15)(x – 1)

(b) (4x – 15)(x + 1)

(c) (4x – 1)(x + 15)

(d) (4x + 1)(x – 15)

(e) (4x + 5)(x – 3)

(f) (4x – 5)(x + 3)

(g) (4x + 3)(x – 5)

(h) (4x – 3)(x + 5)

(i) (2x + 15)(2x – 1)

(j) (2x – 15)(2x + 1)

(k) (2x + 5)(2x – 3)

(l) (2x – 5)(2x + 3)

We have chosen these combinations to force the first and last terms of the quadratic to be 4 x 2 and –15, respectively, we only need to check the combination that will give a middle term of –4 x (if there is one).

(a) –4x + 15x = 11x

(b) 4x – 15x = –11x

(c) 60xx = 59x

(d) –60x + x = –59x

(e) –12x + 5x = –7x

(f) 12x – 5x = 7x

(g) –20x + 3x = –17x

(h) 20x – 3x = 17x

(i) –2x + 30x = 28x

(j) 2x – 30x = –28x

(k) –6 x + 10 x = 4x

(l) 6x – 10x = –4x

Combination (l) is the correct factorization:

4x2 – 4x – 15 = (2x – 5)(2x + 3).

You can see that when the constant term and x2 ’s coefficient have many factors, this list of factorizations to check can grow rather long. Fortunately there is a way around this problem as we shall see in a later chapter.

Find practice problems and solutions at Factoring Quadratic Polynomials Practice Problems - Set 7.

More practice problems for this concept can be found at: Algebra Factoring Practice Test.

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