# Algebraic Expressions

## Exercise 9.4

Question 1: Multiply the binomials.

(i) (2x + 5) and (4x – 3)

Answer: (2x + 5)(4x - 3)
= 2x × 4x - 2x × 3 + 5 × 4x - 5 × 3
= 8x^2 - 6x + 20x -15
= 8x^2 + 14x -15

(ii) (y – 8) and (3y – 4)

Answer: ( y - 8)(3y - 4)
= y × 3y - 4y - 8 × 3y + 32
= 3y^2 - 4y - 24y + 32
= 3y^2 - 28y + 32

(iii) (2.5l – 0.5m) and (2.5l + 0.5m)

Answer: (2.5l - 0.5m)(2.5l + 0.5)
Using (a+b)(a-b) = a^2 - b^2
We get = 6.25 l^2 - 0.25 m^2

(iv) (a + 3b) and (x + 5)

Answer: = ax + 5a + 3bx + 15b

(v)(2pq + 3q^2) and (3pq – 2q^2)

Answer: = 2pq × 3pq - 2pq × 2q^2 + 3q^2 × 3pq - 3q^2 × 2q^2
= 6p^2q^2 - 4pq^3 + 9pq^3 - 6q^4
= 6p^2q^2 - 5pq^3 - 6p^4

(vi) (3/4a^2+3b^2)xx4(a^2-2/3b^2)

Answer: =3/4a^2\xx4a^2-3/4a^2\xx8/3b^2+12a^2b^2-8b^2

=3a^4-2a^2b^2+12a^2b^2-8b^4

=3a^4-8b^2+10a^2b^2

Question 2: Find the product.

(i) (5 – 2x) (3 + x)

Answer: = 15 + 5x - 6x - 2x^2
= 15 - x - x^2

(ii) (x + 7y) (7x – y)

Answer: = 7x^2 - xy + 49xy - 7y^2
= 7x^2 - 7y^2 + 48xy

iii) (a^2+ b) (a + b^2)

Answer: a^2 xx\ a + a^2\xx\b + a\ xx\ b + b^3
= a^3 + a^2b + ab + b^3
= a^3 + b^3 + a^2b + ab

(iv) (p^2– q^2) (2p + q)

Answer: = 2p^3 + p^2q - 2pq^2 - q^3
= 2p^3 - q^3 + p^2q - 2pq^2

Question 3: Simplify.

(i) (x^2– 5) (x + 5) + 25

Answer: = x^3 + 5x^2 - 5x - 25 + 25
= x^3 + 5x^2 -5x

(ii) (a^2+ 5) (b^3+ 3) + 5

Answer: a^2b^3 + 3a^2 + 5b^3 + 15 + 5
= a^2b^3 + 5b^3 + 3a^2 + 20

(iv) (a + b) (c – d) + (a – b) (c + d) + 2 (ac + bd)

Answer: = (ac - ad + bc - bd) + (ac + ad - bc - bd) + (2ac + 2bd)
= ac - ad + bc - bd + ac + ad - bc - bd + 2ac + 2bd= 4ac

(v) (x + y)(2x + y) + (x + 2y)(x – y)

Answer: =2x^2 + xy + 2xy + y^2 + x^2 - xy + 2xy - 2y^2
= 3x^2 + 4xy - y^2

(vi) (x + y)(x^2– xy + y^2)

Answer: = x^3 - x^2y + xy^2 + x^2y - xy^2 + y^3
= x^3 + y^3

(vii) (1.5x – 4y)(1.5x + 4y + 3) – 4.5x + 12y

Answer: = 2.25x^2 + 6xy + 4.5x - 6xy - 16y^2 - 12y - 4.5x + 12y
= 2.25x^2 - 16y^2

(viii) (a + b + c)(a + b – c)

Answer: = a^2 + ab - ac + ab + b^2 - bc + ac + bc - c^2
= a^2 + b^2 - c^2 + 2ab