Exercise 3.4 - Chapter 3 Algebra 10th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Ex $3.4$
Question 1.
Reduce each of the following rational expressions to its lowest form.
(i) $\frac{x^{2}-1}{x^{2}+x}$
(ii) $\frac{x^{2}-11 x+18}{x^{2}-4 x+4}$
(iii) $\frac{9 x^{2}+81 x}{x^{3}+8 x^{2}-9 x}$
(iv) $\frac{p^{2}-3 p-40}{2 p^{3}-24 p^{2}+64 p}$

Solution:
(i) $\frac{x^{2}-1}{x^{2}+x}=\frac{(x+1)(x-1)}{x(x+1)}=\frac{x-1}{x}$
(ii) $\frac{x^{2}-11 x+18}{x^{2}-4 x+4}=\frac{(x-2)(x-9)}{(x-2)(x-2)}=\frac{x-9}{x-2}$
(iii) $\frac{9 x^{2}+81 x}{x^{3}+8 x^{2}-9 x}=\frac{9 x(x+9)}{x\left(x^{2}+8 x-9\right)}=\frac{9(x+9)}{(x+9)(x-1)}$
$=\frac{9}{x-1}$
(iv)
$\begin{aligned}
\frac{p^{2}-3 p-40}{2 p^{3}-24 p^{2}+64 p} &=\frac{(p-8)(p+5)}{2 p\left(p^{2}-12 p+32\right)} \\
&=\frac{(p-8)(p+5)}{2 p(p-4)(p-8)} \\
&=\frac{p+5}{2 p(p-4)}
\end{aligned}$

Question $2 .$
Find the excluded values, if any of the following expressions.
(i) $\frac{y}{y^{2}-25}$
(ii) $\frac{t}{t^{2}-5 t+6}$
(iii) $\frac{x^{2}+6 x+8}{x^{2}+x-2}$
(iv) $\frac{x^{3}-27}{x^{3}+x^{2}-6 x}$

Solution:
(i) $\frac{y}{y^{2}-25}=\frac{y}{(y+5)(y-5)}$ is undefined when $(\mathrm{y}+5)(\mathrm{y}-5)=0$ that is $\mathrm{y}=-5,5$
$\therefore$ The excluded values are $-5,5$
(ii) $\frac{t}{t^{2}-5 t+6}$ is undefined when $t^{2}-5 t+6=0$ i.e.
$(t-3)(t-2)=0 \Rightarrow t=3,2$
$\therefore$ The excluded values are 3,2
(iii) $\frac{x^{2}+6 x+8}{x^{2}+x-2}$ is undefined when $x^{2}+x-2=0$ i.e.
$(x+2)(x+1)=0$
$\therefore$ The excluded values are 2,1
(iv) $\frac{x^{5}-27}{x^{3}+x^{2}-6 x}$ is undefined when $x^{3}+x^{2}-6 x=0$, i.e
$\begin{aligned}
&x\left(x^{2}+x-6\right)=0 \\
&x(x+3)(x-2)=0
\end{aligned}$
$\therefore$ The excluded values are $-3,2$