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In the package on Factorizing Expressions we looked at how to
factorize quadratic expressions which have the number 1 in front
of the highest order term, x showing that the coeffcient of the square term, x This is the information needed to find the factors of quadratic expressions. Example 1 Factorize the following expressions. (a) 2x Solution (a) The factors of 2 are 2 and 1, and the factors of 3 are 3 and 1. If the quadratic expression factorizes then it is likely to be of the form (2x + c)(1x + d) and the choice for c, d is 3, 1 or 1, 3. Trying the first combination, (2x + 3)(x + 1) = 2x = 2x The second choice is (2x + 1)(x + 3) = 2x = 2x (b) There is more than one choice for the first term since 10 is 1 × 10 as well as 2 × 5. The final term will factor as 2 × 1. Which combination of pairs, either (1, 10) with (2, 1), or (2, 5) with (2, 1), will give the correct coeffcient of x, i.e., 9? The latter two pairs seem the more likely since 2 × 2 + 5 × 1 = 9. Checking (2x + 1)(5x + 2) = 10x = 10x Exercise 1. Factorize each of the following expressions. (a) 2x (b) 3x (c) 3y (d) 4z (e) 64z (f) 4w Solution (a) In this case we have 2x (b) In this case we have 3x (c) In this case we have 3y (d) In this case we have 4z (e) In this case we have 64z (f) This is a case of the difference of two squares which was
seen in the package on Brackets. 4w Quiz To which of the following does 12x2 + 17x - 14 factorize? (a) (12x + 7)(x - 2) (b) (x + 2)(12x - 7) (c) (4x + 7)(x - 3) (d) (x - 7)(4x + 3) Solution There are several possibilities since the final term is -14
and the two quantities corresponding to c and d must therefore
have opposite signs. The possible factors of 12 are (1, 12), (2,
6), (3, 4). For -14, the possible factors are (±1, (1, 12) and (±1, (2, 6) and (±1, (3, 4) and (±1, By inspection (2 × 12) + (1 × {-7}) = 24 - 7 = 17, so the factors appear to be (x + 2) and (12x - 7). This can easily be checked. (x + 2)(12x - 7) = 12x = 12x and the required factorization has been achieved. |