## 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.

- 4x
^{2}– 3x + 7 - y
^{2}+ √2 - 3√t + t√2
- y + \(\frac{2}{y}\)
- x
^{10}+ y^{3}+ t^{50}.

Solution:

1. 4x^{2} – 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. y^{2} + √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. x^{10} + y^{3} + t^{50}

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

Given a polynomial function, identify the degree and leading coefficient calculator…. the following exercises, graph the polynomial functions using a calculator.

Question 2.

Write the coefficients of x2 in each of the following:

- 2 + x
^{2}+ x - 2 – x
^{2}+ x^{3} - \(\frac{π}{2}\)x
^{2}+ x - \(\sqrt{2x}\) – 1

Solution:

1. 2 + x^{2} + x

Coefficient of x^{2} = 1

2. 2 – x^{2} + x^{3}

Coefficient of x^{2} = – 1

3. \(\frac{π}{2}\)x^{2} + x

Coefficient of x^{2} = \(\frac{π}{2}\)

4. \(\sqrt{2x}\) – 1

Coefficient of x^{2} = 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 3x^{35} – 4.

One example of a monomial of degree 100 is √2y^{100}

Question 4.

Write the degree of each of the following polynomials:

- 5x
^{3}+ 4x^{2}+ 7x - 4 – y
^{2} - 5x
^{3}– √7 - 3

Solution:

1. 5x^{3} + 4x^{2} + 7x

Term with the highest power of x = 5x^{3}

Exponent of x in this term = 3

∴ Degree of this polynomial = 3.

2. 4 – y^{2}

Term with the highest power of y = – y^{2}

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.

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Question 5.

Classify the following as linear, quadratic and cubic polynomials:

- x
^{2}+ x - x – x
^{3} - y + y
^{2}+ 4 - 1 + x
- 3t
- r
^{2} - 7x
^{3}

Solution:

- Quadratic
- Cubic
- Quadratic
- Linear
- Linear
- Quadratic
- 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:

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