Problem-1 If the values of a – b and ab are 6 and 40 respectively, find the values of a2 + b2 and (a + b)2.
Solution: a2 + b2 = (a – b)2 + 2ab = 62 + 2(40) = 36 + 80 = 116
(a + b)2 = (a – b)2 + 4ab = 62 + 4(40)= 36 + 160 = 196
Problem-2. If (x + p)(x + q) = x2 – 5x – 300, find the value of p2 + q2.
Solution:
By product formula, we have (x + p) (x + q) = x2 + (p + q)x + pq.
So, by comparison, we get p + q = –5, pq = –300.
Now, we have p2 + q2 = (p + q)2 – 2 pq = (–5)2 –2(–300) = 25 + 600 = 625.
Problem-3. If (x + a)(x + b)(x + c) ≡ x3 – 6x2 + 11x – 6, find the value of a2 + b2 + c2.
Solution: From the product formula, we have
(x+a)(x+b)(x+c) = x3 + (a + b + c)x2 + (ab + bc + ca)x + abc.
Comparing, we get a + b + c = –6, ab + bc + ca = 11, abc = –6.
∴ a2 + b2 + c2 = (a + b + c)2 –2 (ab + bc + ca) = (– 6)2 – 2(11) = 36 – 22 = 14.
Problem-4 If a+b=2 and a2+b2=8,find a3+b3 and a4+b4.
Solution:
a2 + 2ab + b2= (a+b)2
2ab = (a+b)2 – (a2 + b2) = (2)2 – (8) = 4 – 8 = – 4. ∴ ab = 21(2ab) = 21(–4) = – 2.
a3+ b3 = (a+b)3 – 3ab(a+b) = (2)3 – 3(–2)(2) = 8 – 3 (– 4) = 8 + 12 = 20.
Problem-5 . Factorize (x – y)3 + (y – z)3 + (z – x)3.
Solution: Put X = x – y, Y = y – z, Z = z – x. Then
X + Y + Z = (x – y) + (y – z) + (z – x) = x – y + y – z + z – x = 0.
∴ X 3 + Y 3 + Z 3 = 3XYZ
Substituting for X, Y and Z
(x – y)3 + (y – z)3 + (z –x)3 = 3(x – y) (y – z) (z – x).
Problem-6. Factorize x4 +1.
Solution: Adding and subtracting 2x2, the given expression is
x4 + 1 = (x4 + 2x2 + 1) – 2x2
= [(x2)2 + 2(x2)(1) + (1)2] – 2x2
= (x2 + 1)2 – ( 2x)2
= [(x2 + 1) + (2x)][(x2 + 1) – (2x)]
= (x2 + 2x + 1) (x2 – 2x + 1).
Problem-7 Factorize x4 + x2y2 + y4.
Solution: Adding and subtracting x2y2, we get
x4 + x2y2 + y4 = (x4 + 2x2y2 + y4) – x2y2
= [(x2)2 + 2(x2)(y2) + (y2)2] – (xy)2
= (x2 + y2)2 – (xy)2
= [x2 + y2 + (xy)] [x2+y2 – (xy)]
= (x2 + xy + y2) (x2 – xy + y2).
Problem-8 Factorize x4 + 5x2 + 9.
Solution : Adding and subtracting x2, we get
x4 + 5x2 + 9 = (x4 + 6x2 + 9) – x2
= (x2 + 3)2 – x2 = [(x2 + 3) + x][(x2+3) – x]
= (x2 + x + 3) (x2 – x + 3).
Do yourself:
1. If a + b = 5 and a – b = 4, find a2 + b2 and ab.
2. If a + b = 10 and ab = 20, find a2 + b2 and (a–b)2.
3. If (x + l)(x + m) = x2 + 4x + 2, find l2 + m2 and (l – m)2.
4. If a + b + c = 11 and ab + bc + ca = 38, find a2 + b2 + c2.
5. If (x + a)(x + b)(x + c) ≡ x3 – 9x2 + 23x – 15, find a + b + c, 1/a + 1/b + 1/c and a2 + b2 + c2
6. If 2a – 3b = 2 and ab = 6, find 8a3 – 27b3.
7. If ,31=+xx find x2 +21x and x3+ 31x.
8. . If x + y = 6 and xy = 8, find x2 + y2 and x3 + y3.
9. . If p + q = 6 and p2 + q2 = 32, find pq, p3 + q3 and p4 + q4.
Factorization formulae
(ii) (X – Y)2 = X 2 – 2XY + Y 2
(iii) (X + Y)(X–Y) = X2 – Y 2
(iv) ( X + Y) (X 2 – XY + Y 2) = X 3 + Y 3
(v) (X – Y)(X 2 + XY + Y 2) = X 3 – Y 3
(vi) (X + Y)3 = X 3 + Y 3 + 3X 2Y + 3XY 2 = X 3 + Y 3 + 3X Y (X +Y )
(vii) (X – Y)3 = X 3 – Y 3 – 3X 2Y + 3XY 2 = X 3 – Y 3 – 3X Y (X –Y )
(viii) (X + Y + Z)2 = X 2 + Y 2 + Z 2 + 2XY + 2YZ + 2ZX
(ix) (X + Y + Z) (X 2 + Y 2 +Z 2 – XY – YZ – ZX) = X 3 + Y 3 + Z 3 – 3XYZ
Factorize the polynomial using the factorization formulae:
10. 1 + 6x + 9x2
11. 144x2 – 72x + 9
12. 4a2b2 + 20abcd + 25c2d2
13. x2 + y2 – a2 – b2 + 2xy + 2ab
14. 33x3y3 + 27z3
15. (x + y)3 + 8y3
16. (x2 + 1)3 + (x2 – 1)3
17. x6 – y6
14. (x+y)3 − (x–y)3
15. (p+q)3 + (p–q)3 + 6p (p2 – q2)
16. 27x3 + y3 + 27x2y + 9xy2
17. x3 – 12x2 +48x – 64
18. 8x3 – 27y3 – 36x2y + 54xy2
19. 4x2 + 9y2 + z2 + 12xy + 4xz + 6yz
20. a2 + b2 + 9c2 + 2ab – 6ac – 6bc
21. x3 – y3 + 1 + 3xy
22. 8x3 – 125y3 + 180xy +216
23. 8x3 – 27y3 +z3+18xyz
24. 33a3 – 8b3 – 125c3 − 30 3abc
25. (a –2b)3 + (2b – 3c)3 + (3c – a)3
26. (x + y – 2z)3 + (y + z – 2x)3 + (z + x – 2y)3
27. (a2 – b2)3 + (b2 – c2) 3 + (c2 – a2)3
28. a3(b – c)3 + b3(c – a)3 + c3(a – b)3
29. x(x + z) – y(y + z)
30. 1–2xy–x2 – y2
31. x4 +4
Factorization of the quadratic expression ax2 + bx + c
Resolve into factors each of the following:
1. x2 + 7x +12
2. x2 + 9x + 20
3. d2 + 10d +21
4. z2 – 7z – 98
5. a2 – a –72
6. x2 + x – 90
7. p2 – 8p +15
8. y2 – 13y + 42
9. y2 – 20y + 99
10. t2 – 28t + 195
Factorize each of the following:
11. 2a2 + 13a + 15
12. 4x2 + 8x + 3
13. 4x2 + 12x + 9
14. 6x2 + x – 1
15. 6p2 + 17p + 10
16. 4a2 – 11a – 15
17. 7m2 + 16m – 15
18. 8p2 + 29p – 12
19. 6x2 + 5x – 6
20. 15y2 – 13y – 6
21. 14x2 – x – 3
22. 9a2 – 9a + 2
23. 2a2 – 13a + 18
24. 12x2 – 7x + 1
25. 16x2 – 32x +7
Resolve into factors each of the following :
26. 9x2 + 24xy + 15y2
27. 4x2 – 16xy – 9y2
28. 6c2 + 11cd – 10d2
29. 5x2 – 11xy + 6y2
30. 2a2 – 15ab + 28b2
Factorize the following:
31. √2 x2 + 3x +√2
32. √3 x2 + 11x + 6√3
33. 5√5 x2 + 20 x + 3 √5
34. 2x2 + 3√5 x + 5
35. 7x2 + 2√14 x +2
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