Page 127 - Start Up Mathematics_8 (Non CCE)

Step 3: Pair the second and fourth term: 15 + 25p = 5 × 3 + 5 × 5p = 5(3 + 5p)
= 5(5p + 3)
Note that (5p + 3) is a common factor in step 2 and step 3.
Step 4: Combine step 2 and step 3 together.
(15pq + 9q) + (15 + 25p) = 3q(5p + 3) + 5(5p + 3) = (3q + 5)(5p + 3)
Example 6: Factorize the following:
3
2
(a) x(x + y – z) – yz (b) l x + l (x – y) – l(y + z) – z
2
2
2
2
(c) (x – 2x) – 3(x – 2x) –y(x – 2x) + 3y
2
Solution: (a) x(x + y – z) – yz = x + xy – xz – yz (Simplifying the expression)
2
= (x + xy) – (xz + yz) (Regrouping)
= x(x + y) – z(x + y)
= (x – z)(x + y) {Taking (x + y) common}
3
3
2
2
2
(b) l x + l (x – y) – l(y + z) – z = l x + l x – l y – ly – lz – z (Simplifying the expression)
2
2
3
= (l x + l x) – (l y + ly) – (lz + z) (Regrouping)
2
= l x(l + 1) – ly(l + 1) – z(l + 1)
2
= (l x – ly – z)(l + 1) {Taking (l + 1) common}
2
2
2
2
(c) (x – 2x) – 3(x –2x) – y(x – 2x) + 3y
2
2
2
2
= {(x – 2x) – 3(x – 2x)} – {y(x – 2x) – 3y} (Regrouping)
2
2
2
= (x – 2x)(x – 2x – 3) – y(x – 2x – 3)
2
2
2
= (x – 2x – y)(x – 2x – 3) {Taking (x – 2x – 3) common}
EXERCISE 7.2
Factorize the following:
2
2
(a) ab – cb + ad – cd (b) 6xy – y + 12xz – 2yz (c) b – ab(1 – a) – a 3
2
3
2
2
2
2
2
(d) a – a(x + 2y) + 2xy (e) x – 2x y + 3xy – 6y 3 (f) xy(a + 1) – a(x + y )
2
2
3
(g) 8(p – q) – 12(p – q) 2 (h) (lx – my) + (mx + ly) (i) 1 + xy – x – y
Factorization of Binomial Expressions When Expressed as the Difference of Two Squares
The factorization of binomial expressions expressible as the difference of two squares involves the following
2
2
identity: p – q = (p + q)(p – q)
2
4
4
Example 7: Factorize: (a) 16x – 25y 2 (b) 81x – 625 (c) l – (m + n) 4
2
2
2
Solution: (a) 16x – 25y = (4x) – (5y) 2
2
2
= (4x + 5y)(4x – 5y) {Using p – q = (p + q)(p – q)}
4
2 2
(b) 81x – 625 = (9x ) – (25) 2
2
2
2
2
= (9x + 25)(9x – 25) {Using p – q = (p + q)(p – q)}
2
2
2
= (9x + 25){(3x) – 5 }
2
2
2
= (9x + 25)(3x + 5)(3x – 5) {Using p – q = (p + q)(p – q)}
4
2 2
4
2 2
(c) l – (m + n) = (l ) – {(m + n) }
2
2
2
2
2
2
= {l + (m + n) } {l – (m + n) } {Using p – q = (p + q)(p – q)}
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