Here we will solve different types of Problems on Factorization by grouping of terms.
1. Factorize: a^{2} – (b – 5)a – 5b.
Solution:
Given expression = a^{2} – (b – 5)a – 5b
= a^{2} – ba + 5a – 5b
= a(a - b) + 5(a - b)
= (a – b)(a + 5).
2. Factorize: a^{2} + b^{2} + a + b + 2ab
Solution:
Given expression = a^{2} + b^{2} + a + b + 2ab
= a^{2} + 2ab + b^{2} + (a + b)
= (a + b)^{2} + (a + b)
= (a + b){(a + b) + 1}
= (a + b)(a + b + 1).
3. Find the factors by grouping of terms: x^{2} – 2x – 2y + xy
Solution:
Given expression = x^{2} – 2x – 2y + xy
= x^{2} – 2x + xy – 2y
= x(x – 2) + y(x – 2)
= (x – 2)(x + y).
Therefore, the factors are x – 2 and x + y.
4. Factorize: 5x^{3} – 15x^{2} – x + 3
Solution:
Given expression = 5x^{3} – 15x^{2} – x + 3
= 5x^{2}(x – 3) – 1(x – 3)
= (x – 3)(5x^{2} – 1)
5. Factorize by grouping: x^{2} – (a + 4)x + 4a
Solution:
Given expression = x^{2} – (a + 4)x + 4a
= x^{2} – ax – 4ax + 4a
= x(x – a) – 4a(x – a)
= (x – a)(x – 4a)
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