Chapter 02 Polynomials Exercise-01
EXERCISE 2.1
1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) $4 x^{2}-3 x+7$
(ii) $y^{2}+\sqrt{2}$
(iii) $3 \sqrt{t}+t \sqrt{2}$
(iv) $y+\frac{2}{y}$
(v) $x^{10}+y^{3}+t^{50}$
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Solution
(i) $4 x^{2}-3 x+7$
Yes, this expression is a polynomial in one variable $x$.
(ii) $y^{2}+\sqrt{2}$
Yes, this expression is a polynomial in one variable $y$.
(iii) $3 \sqrt{t}+t \sqrt{2}$
No. It can be observed that the exponent of variable $t$ in term $3 \sqrt{t} \quad \frac{1}{2}$ is, which is not a whole number. Therefore, this expression is not a polynomial.
(iv) $y+\frac{2}{y}$
2. Write the coefficients of $x^{2}$ in each of the following:
(i) $2+x^{2}+x$
(ii) $2-x^{2}+x^{3}$
(iii) $\frac{\pi}{2} x^{2}+x$
(iv) $\sqrt{2} x-1$
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Solution
(i) $2+x^{2}+x$
In the above expression, the coefficient of is 1 . (ii) $2-x^{2}+x^{3}$ $x^{2}$
In the above expression, the coefficient of $x^{2}$ is -1 .
(iii) $\frac{\pi}{2} x^{2}+x$
In the above expression, the coefficient of $x^{2}$ is $^{\frac{\pi}{2}}$.
$ \begin{aligned} & \sqrt{2} x-1 \text{, or } \\ & \text{ (iv) } \\ & 0 . x^{2}+\sqrt{2} x-1 \end{aligned} $
In the above expression, the coefficient of $x^{2}$ is 0 .
3. Give one example each of a binomial of degree 35 , and of a monomial of degree 100.
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Solution
Degree of a polynomial is the highest power of the variable in the polynomial.
Binomial has two terms in it. Therefore, binomial of degree 35 can be written as
$x^{35}+x^{34}$.
Monomial has only one term in it. Therefore, monomial of degree 100 can be written as $x^{100}$.
4. Write the degree of each of the following polynomials:
(i) $5 x^{3}+4 x^{2}+7 x$
(ii) $4-y^{2}$
(iii) $5 t-\sqrt{7}$
(iv) 3
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Solution
Degree of a polynomial is the highest power of the variable in the polynomial.
(i) $5 x^{3}+4 x^{2}+7 x$
This is a polynomial in variable $x$ and the highest power of variable $x$ is 3 . Therefore, the degree of this polynomial is 3 .
(ii)
$ 4-y^{2} $
This is a polynomial in variable $y$ and the highest power of variable $y$ is 2 . Therefore, the degree of this polynomial is 2 .
(iii) $5 t-\sqrt{7}$
This is a polynomial in variable $t$ and the highest power of variable $t$ is 1 . Therefore, the degree of this polynomial is 1 .
(iv) 3
This is a constant polynomial. Degree of a constant polynomial is always 0 .
5. Classify the following as linear, quadratic and cubic polynomials:
(i) $x^{2}+x$
(ii) $x-x^{3}$
(iii) $y+y^{2}+4$
(iv) $1+x$
(v) $3 t$
(vi) $r^{2}$
(vii) $7 x^{3}$
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Solution
Linear polynomial, quadratic polynomial, and cubic polynomial has its degrees as 1, 2, and 3 respectively.
$ \begin{aligned} & x^{2}+x \\ & x-x^{3} \quad \text{ as its degree is } 2 \text{. } \\ & y+y^{2}+4 \\ & \text{ (i) is a quadratic polynomial } \end{aligned} $
(vi) is a quadratic polynomial as its degree is 2 .
(vii) is a cubic polynomial as its degree is 3 .
