Exercise 3.1 - Chapter 3 Algebra 9th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

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On April 24, 2024, 11:35 AM

Ex $3.1$ : Chapter 3 - Algebra - 9th Maths Guide Samacheer Kalvi Solutions
Question 1.
Which of the following expressions are polynomials. If not give reason:
(i) $\frac{1}{x^{2}}+3 x-4$
(ii) $x^{2}(x-1)$
(iii) $\frac{1}{x}(x+5)$
(iv) $\frac{1}{x^{-2}}+\frac{1}{x^{-1}}+7$
(v) $\sqrt{5} x^{2}+\sqrt{3} x+\sqrt{2}$
(vi) $m^{2}-\sqrt[3]{m}+7 m-10$
Solution:

Question $2 .$
Write the coefficient of $x^{2}$ and $x$ in each of the following polynomials,
(i) $4+\frac{2}{5} x^{2}-3 \mathrm{x}$
(ii) $6-2 \mathrm{x}^{2}+3 \mathrm{x}^{3}-\sqrt{7} x$
(iii) $\pi \mathrm{x}^{2}-\mathrm{x}+2$
(iv) $\sqrt{3} x^{2}+\sqrt{2} x+0.5$
(v) $x^{2}-\frac{7}{2} x+8$
Solution:

Question 3 .
Find the degree of the following polynomials.
(i) $\quad 1-\sqrt{2} y^{2}+y^{7}$
(ii) $\frac{x^{3}-x^{4}+6 x^{6}}{x^{2}}$
(iii) $x^{3}\left(x^{2}+x\right)$
(iv) $3 x^{4}+9 x^{2}+27 x^{6}$
(v) $2 \sqrt{5} p^{4}-\frac{8 p^{3}}{\sqrt{3}}+\frac{2 p^{2}}{7}$

Solution:

Question $4 .$
Rewrite the following polynomial in standard form.
(i) $x-9+\sqrt{7} x^{3}+6 x^{2}$
(ii) $\sqrt{2} x^{2}-\frac{7}{2} x^{4}+x-5 x^{3}$
(iii) $7 x^{3}-\frac{6}{5} x^{2}+4 x-1$
(iv) $y^{2}+\sqrt{5} y^{3}-11-\frac{7}{3} y+9 y^{4}$

Solution:

Question $5 .$
Add the following polynomials and find the degree of the resultant polynomial.
(i) $p(x)=6 x^{2}-7 x+2 q(x)=6 x^{3}-7 x+15$
(ii) $h(x)=7 x^{3}-6 x+1 f(x)=7 x^{2}+17 x-9$
(iii) $f(x)=16 x^{4}-5 x^{2}+9 g(x)=-6 x^{3}+7 x-15$

Solution:

Question $6 .$
Subtract the second polynomial from the first polynomial and find the degree of the resultant polynomial.
(i) $\mathrm{p}(\mathrm{x})=7 \mathrm{x}^{2}+6 \mathrm{x}-1 \mathrm{q}(\mathrm{x})=6 \mathrm{x}-9$
(ii) $f(y)=6 y^{2}-7 y+2 g(y)=7 y+y^{3}$
(iii) $h(z)=z^{5}-6 z^{4}+z f(z)=6 z^{2}+10 z-7$

Solution:

Question $7 .$
What should be added to $2 x^{3}+6 x^{2}-5 x+8$ to get $3 x^{3}-2 x^{2}+6 x+15$ ?
Solution:
$\begin{aligned}
&\left(2 x^{3}+6 x^{2}-5 x+8\right)+Q(x)=3 x^{3}-2 x^{2}+6 x+15 \\
&\therefore Q(x)=\left(3 x^{3}-2 x^{2}+6 x+15\right)-\left(2 x^{3}+6 x^{2}-5 x+8\right) \\
&\quad 3 x^{3}-2 x^{2}+6 x+15
\end{aligned}$
$\frac{(-)_{(\rightarrow)}^{\left.2 x^{3}+6 x^{2}-5\right)} 5 x+8}{x^{3}-8 x^{2}+11 x+7}$
The required polynomial is $x^{3}-8 x^{2}+11 x+7$
 

Question 8.
What must be subtracted from $2 x^{24}+4 x^{2}-3 x+7$ to get $3 x^{3}-x^{2}+2 x+1$ ?
Solution:
$\begin{aligned}
&\left(2 x^{4}+4 x^{2}-3 x+7\right)-Q(x)=3 x^{3}-x^{2}+2 x+1 \\
&Q(x)=\left(2 x^{4}+4 x^{2}-3 x+7\right)-3 x^{3}-x^{2}+2 x+1
\end{aligned}$
The required polynomial $=2 x^{4}+4 x^{2}-3 x+7-3 x^{3}+x^{2}-2 x-1$ $=2 x^{4}-3 x^{3}+5 x^{2}-5 x+6$

Question $9 .$
Multiply the following polynomials and find the degree of the resultant polynomial:
(i) $\mathrm{p}(\mathrm{x})=\mathrm{x}^{2}-9 \mathrm{q}(\mathrm{x})=6 \mathrm{x}^{2}+7 \mathrm{x}-2$
(ii) $f(x)=7 x+2 g(x)=15 x-9$
(iii) $h(x)=6 x^{2}-7 x+1 f(x)=5 x-7$
Solution:
(i) $p(x)=x^{2}-9 q(x)=6 x^{2}+7 x-2$
$\mathrm{p}(\mathrm{x}) \times \mathrm{q}(\mathrm{x})=\left(\mathrm{x}^{2}-9\right)\left(6 \mathrm{x}^{2}+7 \mathrm{x}-2\right)$

The required polynomial is $6 x^{4}+7 x^{3}-56 x^{2}-63 x+18$, degree 4
(ii) $f(x)=7 x+2 g(x)=15 x-9$
$f(x) \times g(x)=(7 x+2)(15 x-9)$
The required polynomial is $105 x^{2}-33 x-18$, degree 2 .
(iii) $\mathrm{h}(\mathrm{x})=6 \mathrm{x}^{2}-7 \mathrm{x}+1 \mathrm{f}(\mathrm{x})=5 \mathrm{x}-7$
$h(x) \times f(x)=\left(6 x^{2}-7 x+1\right)(5 x-7)$

The required polynomial is $30 x^{3}-77 x^{2}+54 x-7$, degree 3 .
 

Question $10 .$
The cost of chocolate is Rs. $(x+y)$ and Amir bought $(x+y)$ chocolates. Find the total amount paid by him in terms of $x$ andy. If $x=10, y=5$ find the amount paid by him.
Solution:
Amount paid $=$ Number of chocolates $\times$ Cost of a chocolate
$=(x+y)(x+y)=(x+y)^{2}=x^{2}+2 x y+y^{2}$
If $x=10, y=5$
The total amount paid by him
$=10^{2}+2 \times 10 \times 5+5^{2}=100+100+25=\text { Rs. } 225$

Question $11 .$
The length of a rectangle is $(3 x+2)$ units and it's breadth is $(3 x-2)$ units. Find its area in terms of $x$. What will be the area if $x=20$ units.
Solution:
Area of a rectangle $=$ length $\times$ breadth
$=(3 x+2) \times(3 x-2)=(3 x)^{2}-2^{2}=\left[9 x^{2}-4\right]$ Sq. units
If $x=20$, Area $=9 \times 20^{2}-4=9 \times 400-4$
$=3600-4=3596 \mathrm{Sq}$. units

Question $12 .$
$\mathrm{p}(\mathrm{x})$ is a polynomial of degree 1 and $q(x)$ is a polynomial of degree 2 . What kind of the polynomial $p(x)$ $\times q(x)$ is ?
Solution:
$\mathrm{p}(\mathrm{x})$ is a polynomial of degree $1 . \mathrm{q}(\mathrm{x})$ is a polynomial of degree 2 .
Then the $p(x) \times q(x)$ will be the polynomial of degree $(1+2)=3$ (or)
Cubic polynomial