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🔢 Chapter 4: Exploring Algebraic Identities • Exercise Set 4.5

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Exercise Set 4.1 2Exercise Set 4.2 2Exercise Set 4.3 1Exercise Set 4.4 2Exercise Set 4.5 1End-of-Chapter Exercises 6All Exercises 14
Q1

Exercise Set 4.5, Q1: Simplify the following rational expressions assuming that the expressions in the denominators are not equal to zero:
(i) $\frac{3p^2 - 3pq - 18q^2}{p^2 + 3pq - 10q^2}$
(ii) $\frac{n^3 - 3n^2m + 3nm^2 - m^3}{5m^2 - 10mn + 5n^2}$
(iv) $\frac{4y^2 - 20yz + 25z^2}{25z^2 - 4y^2}$
(v) $\frac{(x^2 + x - 6)(x^2 - 7x + 12)}{(x^2 - 6x + 8)(x^2 - 9)}$
(vi) $\frac{p^4 - 16}{p^2 - 4p + 4}$

Answer:
(i) $\frac{3(p^2 - pq - 6q^2)}{(p+5q)(p-2q)} = \frac{3(p-3q)(p+2q)}{(p+5q)(p-2q)}$
(ii) $\frac{(n-m)^3}{5(m^2 - 2mn + n^2)} = \frac{(n-m)^3}{5(m-n)^2} = \frac{-(m-n)^3}{5(m-n)^2} = \mathbf{\frac{n-m}{5}}$
(iv) $\frac{(2y - 5z)^2}{(5z - 2y)(5z + 2y)} = \frac{(5z - 2y)^2}{(5z - 2y)(5z + 2y)} = \mathbf{\frac{5z - 2y}{5z + 2y}}$
(v) $\frac{(x+3)(x-2)(x-3)(x-4)}{(x-2)(x-4)(x-3)(x+3)} = \mathbf{1}$
(vi) $\frac{(p^2-4)(p^2+4)}{(p-2)^2} = \frac{(p-2)(p+2)(p^2+4)}{(p-2)^2} = \mathbf{\frac{(p+2)(p^2+4)}{p-2}}$