The DAV Class 8 Maths Solutions and DAV Class 8 Maths Chapter 7 Worksheet 5 Solutions of Algebraic Identities offer comprehensive answers to textbook questions.
DAV Class 8 Maths Ch 7 WS 5 Solutions
Question 1.
Find the product by using suitable identity:
(i) (x + 5) (x + 4)
Solution:
(x + 5) (x + 4) = x2 + (5 + 4) x + 5 × 4
= x2 + 9x + 20.
(ii) (a + 3) (a + 6)
Solution:
(a + 3) (a + 6) = a2 + (3 + 6) x + 3 × 6
= a2 + 9x + 18.
(iii) (x – 9) (x + 7)
Solution:
(x – 9) (x + 7) = x2 + (- 9 + 7)x + (- 9) × (7)
= x2 – 2x – 63.
(iv) (x + 8) (x – 5)
Solution:
(x + 8) (x – 5) = x2 + (8 – 5) x + (8) × (- 5)
= x2 + 3x – 40.
(v) (z – 3) (z – 1)
Solution:
(z – 3) (z – 1) = z2 + (- 3 – 1) z + (- 3) × (- 1)
= z2 – 4z + 3.
(vi) (p – 5) (p – 4)
Solution:
(p – 5) (p – 4) = p2 + (- 5 – 4) p + (- 5) × (- 4)
= p2 + 9p + 20.
(vii) (y – 1) (y + 2)
Solution:
(y – 1) (y + 2) = y2 + (- 1 + 2)y + (- 1) × (2)
= y2 + y + 2.
(viii) (z + 3) (z – 7)
Solution:
(z + 3) (z – 7) = z2 + (3 – 7) z + (3) × (- 7)
= z2 – 4z – 21.
(ix) (p + 8) (p – 3)
Solution:
(p + 8) (p – 3) = p2 + (8 – 3) p + (8) × (- 3)
= p2 + 5p – 24.
(x) (z + 6) (z – 5)
Solution:
(z + 6) (z – 5) = z2 + (6 – 5) z + (6) × (- 5)
= z2 + z – 30.
(xi) (x – 6) (x – 9)
Solution:
(x – 6) (x – 9) = x2 + (- 6 – 9) x + (- 6) × (- 9)
= x2 – 15x + 54.
(xii) (x – 10) (x + 9)
Solution:
(x – 10) (x + 9) = x2 + (- 10 + 9) x + (- 10) × (9)
= x2 – x – 90.
(xiii) (y – 4) (y + 4)
Solution:
(y – 4) (y + 4) = (y)2 – (4)2
= y2 – 16
(xiv) (x – 4) (x – 14)
Solution:
(x – 4) (x – 14) = x2 + (- 4 – 14) x + (- 4) × (- 14)
= x2 – 18x + 56.
(xv) (x – 8) (x – 2)
Solution:
(x – 8) (x – 2) = x2 + (- 8 – 2) x + (- 8) × (- 2)
= x2 – 10x + 16.
Question 2.
By using a suitable identity, evaluate the following:
(i) 102 × 104
Solution:
102 × 104 = (100 + 2) (100 + 4)
= (100)2 + (2 + 4) × 100 + 2 × 4
= 10000 + 600 + 8
= 10608.
(ii) 105 × 103
Solution:
105 × 103 = (100 + 5) (100 + 3)
= (100)2 + (5 + 3) × 100 + 5 × 3
= 10000 + 800 + 15
= 10815.
(iii) 206 × 205
Solution:
206 × 205 = (200 + 6X200 + 5)
= (200)2 + (6 + 5) × 200 + 6 × 5
= 40000 + 2200 + 30
= 42230
(iv) 98 × 96
Solution:
98 × 96 = (100 – 2X100 – 4)
= (100)2 + (- 2 – 4) × 100 + (- 2) × (- 4)
= 10000 – 600 + 8
= 9408.
(v) 87 × 85
Solution:
87 × 85 = (80 + 7) (80 + 5)
= (80)2 + (7 + 5) × 80 + 7 × 5
= 6400 + 960 + 35
= 7395
(vi) 104 × 95
Solution:
104 × 95 = (100 + 4) (100 – 5)
= (100)2 + (4 – 5) × 100 + 4 × (- 5)
= 10000 – 100 – 20
= 9880.
(vii) 97 × 102
Solution:
97 × 102 = (100 – 3) (100 + 2)
= (100)2 + (- 3 + 2) × 100 + (- 3) × (2)
= 10000 – 100 – 6
= 9894.
(viii) 203 × 198
Solution:
203 × 198 = (200 + 3) (200 – 2)
= (200)2 + (3 – 2) × 200 + (3) × (- 2)
= 40000 + 200 – 6
= 40194.
(ix) 35 × 37
Solution:
35 × 37 = (40 – 5) (40 – 3)
= (40)2 + (- 5 – 3) × 40 + (- 5) × (- 3)
= 1600 – 320 + 15
= 1295.
(x) 106 × 93
Solution:
106 × 93 = (100 + 6) (100 – 7)
= (1002 + (6 – 7) × 100 + (6) × (- 7)
= 10000 – 100 – 42
= 9858.
Question 3.
Evaluate the following products:
(i) (x2 + 3) (x2 + 4)
Solution:
(x2)2 + (3 + 4) x2 + 3 × 4
= x4 + 7x2 + 12
(ii) (x + \(\frac{4}{3}\)) (x + \(\frac{1}{3}\))
Solution:
(x + \(\frac{4}{3}\)) (x + \(\frac{1}{3}\)) = x2 + \(\left(\frac{-3}{5}-\frac{1}{2}\right) x+\left(\frac{-3}{5}\right) \times\left(-\frac{1}{2}\right)\)
= x2 + \(\frac{5}{3} x+\frac{4}{9}\)
(iii) (x – \(\frac{3}{5}\)) (x – \(\frac{1}{2}\))
Solution:
(x – \(\frac{3}{5}\)) (x – \(\frac{1}{2}\)) = x2 + \(\left(\frac{-3}{5}-\frac{1}{2}\right) x+\left(\frac{-3}{5}\right) \times\left(-\frac{1}{2}\right)\)
= x2 – \(\frac{11}{10} x+\frac{3}{10}\)
(iv) (y2 – 6) (y2 + 7)
Solution:
(y2 – 6) (y2 + 7) = (y2)2 + (- 6 + 7) y2 + (- 6) × (7)
= y4 + y2 – 42
(v) (z2 + 4) (z2 – \(\frac{1}{4}\))
Solution:
(z2 + 4) (z2 – \(\frac{1}{4}\)) = (z2)2 + (4 – \(\frac{1}{4}\)) z2 + (4) (- \(\frac{1}{4}\))
= z4 – \(\frac{15}{4}\)z2 – 1
(vi) (y2 – 3) (y2 – 1)
Solution:
(y2 – 3) (y2 – 1) = (y2)2 + (- 3 – 1) y2 + (- 3) × (- 1)
= y4 + 4y2 + 3
(vii) (x3 + 5) (x3 + 2)
Solution:
(x3 + 5) (x3 + 2) = (x3)2 + (5 + 2) x3 + 5 × 2
= x6 + 7x3 + 10
(viii) (p2 – \(\frac{1}{4}\)) (p2 + \(\frac{1}{8}\))
Solution:
(p2 – \(\frac{1}{4}\)) (p2 + \(\frac{1}{8}\)) = (p2)2 + \(\left(-\frac{1}{4}+\frac{1}{8}\right) p^2+\left(-\frac{1}{4}\right) \times\left(\frac{1}{8}\right)\)
= p4 – \(\frac{1}{8} p^2-\frac{1}{32}\)
(ix) (z + \(\frac{1}{6}\)) (z + 6)
Solution:
(z + \(\frac{1}{6}\)) (z + 6) = z2 + (\(\frac{1}{6}\) + 6) + \(\frac{1}{6}\) × 6
= z2 + \(\frac{37}{6}\) z + 1