1. Which of the following expressions are polynomials in one variable, and which are not? State reasons for your answer.
(i) 4x2–3x+7
(ii) y2+√2
(iii) 3√t+t√2
(iv) y+2/y
(v) x10+y3+t50
(i) 4x2–3x+7
(i) 4x2–3x+7
Ans:
The equation 4x2–3x+7 can be written as 4x2–3x1+7x0
Since x is the only variable in the given equation and the powers of x (i.e. 2, 1 and 0) are whole numbers, we can say that the expression 4x2–3x+7 is a polynomial in one variable.
(ii) y2+√2
Ans:
The equation y2+√2 can be written as y2+√2y0
Since y is the only variable in the given equation and the powers of y (i.e., 2 and 0) are whole numbers, we can say that the expression y2+√2 is a polynomial in one variable.
(iii) 3√t+t√2
Ans:
The equation 3√t+t√2 can be written as 3t1/2+√2t
Though t is the only variable in the given equation, the power of t (i.e., 1/2) is not a whole number. Hence, we can say that the expression 3√t+t√2 is not a polynomial in one variable.
(iv) y+2/y
Ans:
The equation y+2/y can be written as y+2y-1
Though y is the only variable in the given equation, the power of y (i.e., -1) is not a whole number. Hence, we can say that the expression y+2/y is not a polynomial in one variable.
(v) x10+y3+t50
Ans:
Here, in the equation x10+y3+t50
Though the powers, 10, 3, 50, are whole numbers, there are 3 variables used in the expression
x10+y3+t50. Hence, it is not a polynomial in one variable.
2. Write the coefficients of x2in each of the following:
(i) 2+x2+x
(ii) 2–x2+x3
(iii) (π/2)x2+x
(iv)√2x-1
(i) 2+x2+x
Ans:
The equation 2+x2+x can be written as 2+(1)x2+x
We know that the coefficient is the number which multiplies the variable.
Here, the number that multiplies the variable x2 is 1
Hence, the coefficient of x2 in 2+x2+x is 1.
(ii) 2–x2+x3
Ans:
The equation 2–x2+x3 can be written as 2+(–1)x2+x3
We know that the coefficient is the number (along with its sign, i.e. – or +) which multiplies the variable.
Here, the number that multiplies the variable x2 is -1
Hence, the coefficient of x2 in 2–x2+x3 is -1.
(iii) (π/2)x2+x
Ans:
The equation (π/2)x2 +x can be written as (π/2)x2 + x
We know that the coefficient is the number (along with its sign, i.e. – or +) which multiplies the variable.
Here, the number that multiplies the variable x2 is π/2.
Hence, the coefficient of x2 in (π/2)x2 +x is π/2.
(iv)√2x-1
Ans:
The equation √2x-1 can be written as 0x2+√2x-1 [Since 0x2 is 0]
We know that the coefficient is the number (along with its sign, i.e. – or +) which multiplies the variable.
Here, the number that multiplies the variable x2is 0
Hence, the coefficient of x2 in √2x-1 is 0.
3.Give one example each of a binomial of degree 35, and of a monomial of degree 100.
Ans:
Binomial of degree 35: A polynomial having two terms and the highest degree 35 is called a binomial of degree 35.
For example, 3x35+5
Monomial of degree 100: A polynomial having one term and the highest degree 100 is called a monomial of degree 100.
For example, 4x100
4. Write the degree of each of the following polynomials:
(i) 5x3+4x2+7x
(ii) 4–y2
(iii) 5t–√7
(iv) 3
(i) 5x3+4x2+7x
Ans:
The highest power of the variable in a polynomial is the degree of the polynomial.
Here, 5x3+4x2+7x = 5x3+4x2+7x1
The powers of the variable x are: 3, 2, 1
The degree of 5x3+4x2+7x is 3, as 3 is the highest power of x in the equation.
(ii) 4–y2
Ans:
The highest power of the variable in a polynomial is the degree of the polynomial.
Here, in 4–y2,
The power of the variable y is 2
The degree of 4–y2 is 2, as 2 is the highest power of y in the equation.
(iii) 5t–√7
Ans:
The highest power of the variable in a polynomial is the degree of the polynomial.
Here, in 5t–√7
The power of the variable t is: 1
The degree of 5t–√7 is 1, as 1 is the highest power of y in the equation.
(iv) 3
Ans:
The highest power of the variable in a polynomial is the degree of the polynomial.
Here, 3 = 3×1 = 3× x0
The power of the variable here is: 0
Hence, the degree of 3 is 0.
5.Classify the following as linear, quadratic and cubic polynomials:
(i) x2+x
(ii) x–x3
(iii) y+y2+4
(iv) 1+x
(v) 3t
(vi) r2
(vii) 7x3
Ans:
We know that,
Linear polynomial: A polynomial of degree one is called a linear polynomial.
Quadratic polynomial: A polynomial of degree two is called a quadratic polynomial.
Cubic polynomial: A polynomial of degree three is called a cubic polynomial.
(i) x2+x
Ans:
The highest power of x2+x is 2
The degree is 2
Hence, x2+x is a quadratic polynomial
(ii) x–x3
Ans:
The highest power of x–x3 is 3
The degree is 3
Hence, x–x3 is a cubic polynomial
(iii) y+y2+4
Ans:
The highest power of y+y2+4 is 2
The degree is 2
Hence, y+y2+4 is a quadratic polynomial
(iv) 1+x
Ans:
The highest power of 1+x is 1
The degree is 1
Hence, 1+x is a linear polynomial.
(v) 3t
Ans:
The highest power of 3t is 1
The degree is 1
Hence, 3t is a linear polynomial.
(vi) r2
Ans:
The highest power of r2 is 2
The degree is 2
Hence, r2is a quadratic polynomial.
(vii) 7x3
Ans:
The highest power of 7x3 is 3
The degree is 3
Hence, 7x3 is a cubic polynomial.