Exercise 4.5 Practice
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Overview
This page provides comprehensive Ch 4: Algebraic Identities – Exercise 4.5 Practice. Simplify rational expressions by factoring numerators and denominators completely using standard and advanced algebraic identities, with step-by-step solutions.
Simplifying Rational Algebraic Expressions
Q1: Simplify Rational Expressions
Simplify the following rational expressions assuming that the expressions in the denominators are not equal to zero:
(i) $\dfrac{3p^2 - 3pq - 18q^2}{3p^2 - 10pq + 3q^2}$ (ii) $\dfrac{5m^3 - 10m^2n + 5mn^2}{5m^3 - 5n^3}$
(iii) $\dfrac{w^3 - v^3 + x^3 + 3wvx}{w^2 + v^2 + x^2 + wv + vx - wx}$ (iv) $\dfrac{4y^2 - 20yz + 25z^2}{25z^2 - 4y^2}$
(v) $\dfrac{(x^2 - 7x + 12)(x^2 - 9)}{(x^2 - 8x + 12)(x^2 - 6x + 9)}$ (vi) $\dfrac{p^4 - 16}{p^2 - 4p + 4}$
(i) $\dfrac{3p^2 - 3pq - 18q^2}{3p^2 - 10pq + 3q^2}$ (ii) $\dfrac{5m^3 - 10m^2n + 5mn^2}{5m^3 - 5n^3}$
(iii) $\dfrac{w^3 - v^3 + x^3 + 3wvx}{w^2 + v^2 + x^2 + wv + vx - wx}$ (iv) $\dfrac{4y^2 - 20yz + 25z^2}{25z^2 - 4y^2}$
(v) $\dfrac{(x^2 - 7x + 12)(x^2 - 9)}{(x^2 - 8x + 12)(x^2 - 6x + 9)}$ (vi) $\dfrac{p^4 - 16}{p^2 - 4p + 4}$
(i) $\dfrac{3p^2 - 3pq - 18q^2}{3p^2 - 10pq + 3q^2}$:
• Factorise numerator: $3(p^2 - pq - 6q^2) = 3(p - 3q)(p + 2q)$.
• Factorise denominator: $3p^2 - 9pq - pq + 3q^2 = 3p(p - 3q) - q(p - 3q) = (3p - q)(p - 3q)$.
• Simplify:
$$\frac{3(p - 3q)(p + 2q)}{(3p - q)(p - 3q)} = \mathbf{\frac{3(p + 2q)}{3p - q}}$$
• Factorise numerator: $3(p^2 - pq - 6q^2) = 3(p - 3q)(p + 2q)$.
• Factorise denominator: $3p^2 - 9pq - pq + 3q^2 = 3p(p - 3q) - q(p - 3q) = (3p - q)(p - 3q)$.
• Simplify:
$$\frac{3(p - 3q)(p + 2q)}{(3p - q)(p - 3q)} = \mathbf{\frac{3(p + 2q)}{3p - q}}$$
(ii) $\dfrac{5m^3 - 10m^2n + 5mn^2}{5m^3 - 5n^3}$:
• Factorise numerator: $5m(m^2 - 2mn + n^2) = 5m(m - n)^2$.
• Factorise denominator: $5(m^3 - n^3) = 5(m - n)(m^2 + mn + n^2)$.
• Simplify:
$$\frac{5m(m - n)^2}{5(m - n)(m^2 + mn + n^2)} = \mathbf{\frac{m(m - n)}{m^2 + mn + n^2}}$$
• Factorise numerator: $5m(m^2 - 2mn + n^2) = 5m(m - n)^2$.
• Factorise denominator: $5(m^3 - n^3) = 5(m - n)(m^2 + mn + n^2)$.
• Simplify:
$$\frac{5m(m - n)^2}{5(m - n)(m^2 + mn + n^2)} = \mathbf{\frac{m(m - n)}{m^2 + mn + n^2}}$$
(iii) $\dfrac{w^3 - v^3 + x^3 + 3wvx}{w^2 + v^2 + x^2 + wv + vx - wx}$:
• Using the cubic identity: $A^3 + B^3 + C^3 - 3ABC = (A+B+C)(A^2+B^2+C^2 - AB - BC - CA)$.
• Let $A = w$, $B = -v$, $C = x$.
• Numerator $= w^3 - v^3 + x^3 + 3wvx = (w - v + x)(w^2 + v^2 + x^2 + wv + vx - wx)$.
• Simplify:
$$\frac{(w - v + x)(w^2 + v^2 + x^2 + wv + vx - wx)}{w^2 + v^2 + x^2 + wv + vx - wx} = \mathbf{w - v + x}$$
• Using the cubic identity: $A^3 + B^3 + C^3 - 3ABC = (A+B+C)(A^2+B^2+C^2 - AB - BC - CA)$.
• Let $A = w$, $B = -v$, $C = x$.
• Numerator $= w^3 - v^3 + x^3 + 3wvx = (w - v + x)(w^2 + v^2 + x^2 + wv + vx - wx)$.
• Simplify:
$$\frac{(w - v + x)(w^2 + v^2 + x^2 + wv + vx - wx)}{w^2 + v^2 + x^2 + wv + vx - wx} = \mathbf{w - v + x}$$
(iv) $\dfrac{4y^2 - 20yz + 25z^2}{25z^2 - 4y^2}$:
• Numerator $= (2y - 5z)^2$.
• Denominator $= (5z - 2y)(5z + 2y) = -(2y - 5z)(2y + 5z)$.
• Simplify:
$$\frac{(2y - 5z)^2}{-(2y - 5z)(2y + 5z)} = \mathbf{-\frac{2y - 5z}{2y + 5z}} \quad \text{or} \quad \mathbf{\frac{5z - 2y}{2y + 5z}}$$
• Numerator $= (2y - 5z)^2$.
• Denominator $= (5z - 2y)(5z + 2y) = -(2y - 5z)(2y + 5z)$.
• Simplify:
$$\frac{(2y - 5z)^2}{-(2y - 5z)(2y + 5z)} = \mathbf{-\frac{2y - 5z}{2y + 5z}} \quad \text{or} \quad \mathbf{\frac{5z - 2y}{2y + 5z}}$$
(v) $\dfrac{(x^2 - 7x + 12)(x^2 - 9)}{(x^2 - 8x + 12)(x^2 - 6x + 9)}$:
• Factorise each part:
- $x^2 - 7x + 12 = (x - 3)(x - 4)$
- $x^2 - 9 = (x - 3)(x + 3)$
- $x^2 - 8x + 12 = (x - 2)(x - 6)$
- $x^2 - 6x + 9 = (x - 3)^2$
• Substitute and simplify:
$$\frac{(x - 3)(x - 4)(x - 3)(x + 3)}{(x - 2)(x - 6)(x - 3)^2} = \mathbf{\frac{(x - 4)(x + 3)}{(x - 2)(x - 6)}}$$
• Factorise each part:
- $x^2 - 7x + 12 = (x - 3)(x - 4)$
- $x^2 - 9 = (x - 3)(x + 3)$
- $x^2 - 8x + 12 = (x - 2)(x - 6)$
- $x^2 - 6x + 9 = (x - 3)^2$
• Substitute and simplify:
$$\frac{(x - 3)(x - 4)(x - 3)(x + 3)}{(x - 2)(x - 6)(x - 3)^2} = \mathbf{\frac{(x - 4)(x + 3)}{(x - 2)(x - 6)}}$$
(vi) $\dfrac{p^4 - 16}{p^2 - 4p + 4}$:
• Numerator $= (p^2 - 4)(p^2 + 4) = (p - 2)(p + 2)(p^2 + 4)$.
• Denominator $= (p - 2)^2$.
• Simplify:
$$\frac{(p - 2)(p + 2)(p^2 + 4)}{(p - 2)^2} = \mathbf{\frac{(p + 2)(p^2 + 4)}{p - 2}}$$
• Numerator $= (p^2 - 4)(p^2 + 4) = (p - 2)(p + 2)(p^2 + 4)$.
• Denominator $= (p - 2)^2$.
• Simplify:
$$\frac{(p - 2)(p + 2)(p^2 + 4)}{(p - 2)^2} = \mathbf{\frac{(p + 2)(p^2 + 4)}{p - 2}}$$
(i) $\frac{3(p+2q)}{3p-q}$ (ii) $\frac{m(m-n)}{m^2+mn+n^2}$ (iii) $w-v+x$ (iv) $\frac{5z-2y}{2y+5z}$ (v) $\frac{(x-4)(x+3)}{(x-2)(x-6)}$ (vi) $\frac{(p+2)(p^2+4)}{p-2}$