Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i)\(\) 4x2 – 3x + 7
ii)\(\displaystyle y^2 + \sqrt 2\)
iii)\(\displaystyle 3\sqrt t + t\sqrt 2\)
iv)\(\displaystyle y + \dfrac{2}{y}\)
(v) \(\displaystyle x^{10} + y^3 + t^{50}\)
Solution 1:
(i)4x2 – 3x + 7
One variable is involved in given polynomial which is \(x\)
Therefore, it is a polynomial in one variable \(x\).
ii)\(y^2 + \sqrt 2\)
One variable is involved in given polynomial which is \(y\)
Therefore, it 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
\(3\sqrt t\) is \(\dfrac{1}{2}\), which is not a whole number.
Therefore, this expression is not a polynomial.
iv)\(y + \dfrac{2}{y} = y + 2y^{-1}\)
The power of variable ‘\(y\)’ is \(-1\) which is not a whole number.
Therefore, it is not a polynomial in one variable
No. It can be observed that the exponent of variable \(y\) in term
\(\dfrac{2}{y}\) is \(-1\), which is not a whole number.
Therefore, this expression is not a polynomial.
(v) \(x^{10} + y^3 + t^{50}\)
In the given expression there are \(3\) variables which are ‘\(x, y, t\)’ involved.
Therefore, it is not a polynomial in one variable.
Question 2:
Write the coefficients of x2 in each of the following:
(i) \(2 + x^2 + x\)
(ii) \(2 - x^2 + x^3\)
(iii)\(\dfrac{\pi}{2} x^2 + x\)
(iv) \(√2x - 1\)
Solution 2:
(i) \(2 + x^2 + x\)
= \(2 + 1(x^2 )+ x\)
The coefficient of \(x^2\) is \(1\).
(ii) \(2 - x^2 + x^3\)
\(2 - 1(x^2) + x^3\)
The coefficient of \(x^2\) is \(-1\).
(iii)\(\dfrac{\pi}{2} x^2 + x\)
The coefficient \(x^2\) is \(\dfrac{\pi}{2}\).
(iv) \(\sqrt 2x - 1 = 0x^2 + \sqrt 2x - 1\)
The 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 3 :
Binomial of degree \(35\) means a polynomial is having
1.Two terms
2.Highest degree is \(35\)
Example: \(x^{35}+x^{34}\)
Monomial of degree \(100\) means a polynomial is having
1.One term
2.Highest degree is \(100\)
Example : \(x^{100}\).
Question 4:
Write the degree of each of the following polynomials:
(i) \(5x^3 + 4x^2 + 7x\)
(ii) \(4 - y^2\)
(iii) \(5t - \sqrt 7\)
(iv) \(3\)
Solution 4:
Degree of a polynomial is the highest power of the variable in the polynomial.
(i) \(5x^3 + 4x^2 + 7x\)
Highest power of variable ‘\(x\)’ is \(3\). Therefore, the degree of this polynomial is \(3\)
(ii) \(4 - y^2\)
Highest power of variable ‘\(y\)’ is \(2\). Therefore, the degree of this polynomial is \(2\).
(iii) \(5t - \sqrt 7\)
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 \(0\).
Question 5:
Classify the following as linear, quadratic and cubic polynomial:
(ii) x - x3
(iii) y + y2 + 4
(iv) 1 + x
(v) 3t
(vi) r2
(vii) 7x2 + 7x3
Solution 5:
Quadratic polynomial - whose variable highest power is ‘2’
Cubic polynomial- whose variable highest power is ‘3’
(i) x2 + x is a quadratic polynomial as its highest degree is 2.
(ii) x - x3 is a cubic polynomial as its highest degree is 3.
(iii) y + y2 + 4 is a quadratic polynomial as its highest 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) r2 is a quadratic polynomial as its degree is 2.
(vii) 7x2 + 7x3 is a cubic polynomial as highest its degree is 3.
