MP Board Class 9th Maths Solutions Chapter 2 Polynomials Ex 2.1

MP Board Class 9th Maths Solutions Chapter 2 Polynomials Ex 2.1

Question 1.
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.

1. 4x² – 3x + 7
2. y² + √2
3. 3√t + t√2
4. y + \(\frac{2}{y}\)
5. x¹⁰ + y³ + t⁵⁰.

Solution:
1. 4x² – 3x + 7
This expression is a polynomial in one variable .r because in the expression there is only one variable (x) and all the indices of x are whole numbers.

2. y² + √2
This expression is a polynomial in one variable y because in the expression there is only one variable (y) and all the indices of y are whole numbers.

3. 3√t + t√2
This expression is not a polynomial because in the term 3√t, the exponent of t is \(\frac{1}{2}\), which is not a whole number.

4. y + \(\frac{2}{y}\)
This expression is not a polynomial because in the term \(\frac{2}{y}\) the exponent of y is (-1) which is not a whole number.

5. x¹⁰ + y³ + t⁵⁰
This expression is not a polynomial in one variable because in the expression, three variables (x, y and t) occur.

Question 2.
Write the coefficients of x2 in each of the following:

1. 2 + x² + x
2. 2 – x² + x³
3. \(\frac{π}{2}\)x² + x
4. \(\sqrt{2x}\) – 1

Solution:
1. 2 + x² + x
Coefficient of x² = 1

2. 2 – x² + x³
Coefficient of x² = – 1

3. \(\frac{π}{2}\)x² + x
Coefficient of x² = \(\frac{π}{2}\)

4. \(\sqrt{2x}\) – 1
Coefficient of x² = 0.

Question 3.
Give one example each of a binomial of degree 35, and of a monomial of degree 100.
Solution:
One example of a binomial of degree 35 is 3x³⁵ – 4.
One example of a monomial of degree 100 is √2y¹⁰⁰

Question 4.
Write the degree of each of the following polynomials:

1. 5x³ + 4x² + 7x
2. 4 – y²
3. 5x³ – √7
4. 3

Solution:
1. 5x³ + 4x² + 7x
Term with the highest power of x = 5x³
Exponent of x in this term = 3
∴ Degree of this polynomial = 3.

2. 4 – y²
Term with the highest power of y = – y²
Exponent of y in this term = 2
∴ Degree of this polynomial = 2.

3. 5t – √7
Term with the highest power of t = 5t
Exponent of t in this term = 1
∴ Degree of this polynomial = 1.

4. 3
It is a non-zero constant. So the degree of this polynomial is zero.

Question 5.
Classify the following as linear, quadratic and cubic polynomials:

1. x² + x
2. x – x³
3. y + y² + 4
4. 1 + x
5. 3t
6. r²
7. 7x³

Solution:

1. Quadratic
2. Cubic
3. Quadratic
4. Linear
5. Linear
6. Quadratic
7. Cubic.

Zero’s of a Polynomial:
1. A zero of a polynomial p(x) is a number c such that p(c) = 0. Here p(x) = 0 is a polynomial equation and c is the root of the polynomial equation.

2. A non-zero constant polynomial has no zero. Every real number is a zero of the zero polynomial.

Some Observations:

1. A zero of a polynomial need not be 0.
2. 0 may be a zero of a polynomial.
3. Every linear polynomial has one and only one zero.
4. A polynomial can have more than one zero.

MP Board Class 9th Maths Solutions