Here we will learn the process of Factorization of Expressions of the Form x\(^{2}\) + (a + b)x + ab.
We know, (x + a)(x + b) = x\(^{2}\) + (a + b)x + ab.
Therefore, x\(^{2}\) + (a + b)x + ab = (x + a)(x + b).
1. Factorize: a\(^{2}\) + 7a + 12.
Solution:
Here, constant term = 12 = 3 × 4, and 3 + 4 = 7 (= coefficient of a).
Therefore, a\(^{2}\) + 7a + 12 = a\(^{2}\) + 3a + 4a + 12 (breaking 7a is sum of two terms, 3a + 4a)
= (a\(^{2}\) + 3a) + (4a + 12)
= a(a + 3) + 4(a + 3)
= (a + 3)(a + 4).
2. Factorize: m\(^{2}\) – 5m + 6.
Solution:
Here, constant term = 6 = (-2) × (-3), and (-2) + (-3) = -5 (= coefficient of m).
Therefore, m\(^{2}\) – 5m + 6 = m\(^{2}\) -2m – 3m + 6 (breaking -5m is sum of two terms, -2m - 3m)
= (m\(^{2}\) -2m) +(– 3m + 6)
= m(m - 2) - 3(m - 2)
= (m - 2)(m - 3).
3. Factorize: x\(^{2}\)- x - 6.
Solution:
Here, constant term = -6 = (-3) × 2, and (-3) + 2 = -1 (= coefficient of x).
Therefore, x\(^{2}\) - x - 6 = x\(^{2}\) - 3x + 2x - 6 (breaking -x is sum of two terms, -3x + 2x)
= (x\(^{2}\) - 3x) + (2x - 6)
= x(x - 3)+ 2(x - 3)
= (x - 3)(x + 2).
The method of factorizing x\(^{2}\) + px + q by breaking the middle term, as shown in the above examples, involves the following steps.
Steps:
1. Take the constant term (with the sign) q.
2. Break q into two factors, a, b (with suitable signs) whose sum equals the coefficient of x, i.e., a + b = p.
3. Pair one of these, say, ax with x\(^{2}\), and the other, bx, with the constant term q. Then factorize.
Note: In case step 2 is not possible conveniently, x\(^{2}\) + px + q cannot be factorized as above.
For example, x\(^{2}\) + 3x + 4. Here 4 cannot be broken into two factors whose sum is 3.
From Factorization of Expressions of the Form x^2 + (a + b)x + ab to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Dec 12, 24 09:20 AM
Dec 09, 24 10:39 PM
Dec 09, 24 01:08 AM
Dec 08, 24 11:19 PM
Dec 07, 24 03:38 PM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.