Chapter 2 Exercise 2.2
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Question 1:
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
i. $4x^2- 3x + 7$
ii. $y^2 + \sqrt {2}$
iii. $3 \sqrt {t} + t \sqrt { 2}$
iv. $y + \frac {2}{y}$
v. $x^{10} + y^3 + t^{50}$
Solution:
(i)$4x^2- 3x + 7$
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 term3 √t is 1/2, which is not a whole number. Therefore, this expression is not a polynomial.
(iv)$y + \frac {2}{y}$
No. It can be observed that the exponent of variable y in term 2/y is -1 which is not a whole number. Therefore, this expression is not a polynomial.
(v)$x^{10} + y^3 + t^{50}$
No. It can be observed that this expression is a polynomial in 3 variables x, y, and t. Therefore, it is not a polynomial in one variable.
Question 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$
Solution:
(i) $2 + x^2 + x$
Coefficient of $x^2$ is 1.
(ii) $2 - x^2 + x^3$
Coefficient of $x^2$ is -1.
(iii)$(\frac {\pi}{2})x^2 + x$
Coefficient of $x^2$ is (π/2)
(iv) $ \sqrt {2} x -1$
There is no term consisting of $x^2$. Therefore, coefficient of $x^2$ is 0.
Question 3
Give one example each of a binomial of degree 35, and of a monomial of degree 100.
Solution:
Degree of a polynomial is the highest power of variable in the polynomial.
Binomial has two terms in it. So binomial of degree 35 can be written as $x^{35}+ 1$.
monomial has only one term in it. So monomial of degree 100 can be written as $x^{100}$.
Question 4.
Write the degree of each of the following polynomials:
(i) 5x
^{3} + 4x
^{2} + 7x
(ii) 4 - y
^{2}
(iii) 5t -√ 7
(iv) 3
Solution:
(i)
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)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)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)This is a constant polynomial. Degree of a constant polynomial is always 0.
Question 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) 3t
(vi) r
^{2}
(vii) 7x
^{3}
Solution:
(i) 2 + x
^{2}+ x is a quadratic polynomial as its degree is 2.
(ii) x - x
^{3} is a cubic polynomial as its degree is 3.
(iii) y + y
^{2}+ 4 is a quadratic polynomial as its degree is 2.
(iv) 1 + x is a linear polynomial as its degree is 1.
(v) 3t is a linear polynomial as its degree is 1.
(vi) r
^{2}is a quadratic polynomial as its degree is 2.
(vii) 7x
^{3}is a cubic polynomial as its degree is 3.
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