myhelper: RS AGGARWAL CLASS 9 Chapter 2 POLYNOMIALS EXERCISE 2A

EXERCISE 2A

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Question 1:

Which of the following expressions are polynomials? In case of a polynomial, write its degree.
(i) $x^{5} - 2 x^{3} + x + \sqrt{3}$ $x^{5} - 2 x^{3} + x + \sqrt{3}$
(ii) $y^{3} + \sqrt{3} y$ $y^{3} + \sqrt{3} y$
(iii) $t^{2} - \frac{2}{5} t + \sqrt{5}$ $t^{2} - \frac{2}{5} t + \sqrt{5}$
(iv) $x^{100} - 1$ $x^{100} - 1$
(v) $\frac{1}{\sqrt{2}} x^{2} - \sqrt{2} x + 2$ $\frac{1}{\sqrt{2}} x^{2} - \sqrt{2} x + 2$
(vi) $x^{- 2} + 2 x^{- 1} + 3$ $x^{- 2} + 2 x^{- 1} + 3$
(vii) 1
(viii) $\frac{- 3}{5}$ $\frac{- 3}{5}$
(ix) $\frac{x^{2}}{2} - \frac{2}{x^{2}}$ $\frac{x^{2}}{2} - \frac{2}{x^{2}}$
(x) $\sqrt[3]{2} x^{2} - 8$ $\sqrt[3]{2} x^{2} - 8$
(xi) $\frac{1}{2 x^{2}}$ $\frac{1}{2 x^{2}}$
(xii) $\frac{1}{\sqrt{5}} x^{\frac{1}{2}} + 1$ $\frac{1}{\sqrt{5}} x^{\frac{1}{2}} + 1$
(xiii) $\frac{3}{5} x^{2} - \frac{7}{3} x + 9$ $\frac{3}{5} x^{2} - \frac{7}{3} x + 9$
(xiv) $x^{4} - x^{\frac{3}{2}} + x - 3$ $x^{4} - x^{\frac{3}{2}} + x - 3$
(xv) $2 x^{3} + 3 x^{2} + \sqrt{x} - 1$ $2 x^{3} + 3 x^{2} + \sqrt{x} - 1$

Answer 1:

(i) $x^{5} - 2 x^{3} + x + \sqrt{3}$ $x^{5} - 2 x^{3} + x + \sqrt{3}$ is an expression having only non-negative integral powers of x. So, it is a polynomial. Also, the highest power of x is 5, so, it is a polynomial of degree 5.

(ii) $y^{3} + \sqrt{3} y$ $y^{3} + \sqrt{3} y$ is an expression having only non-negative integral powers of y. So, it is a polynomial. Also, the highest power of y is 3, so, it is a polynomial of degree 3.

(iii) $t^{2} - \frac{2}{5} t + \sqrt{5}$ $t^{2} - \frac{2}{5} t + \sqrt{5}$ is an expression having only non-negative integral powers of t. So, it is a polynomial. Also, the highest power of t is 2, so, it is a polynomial of degree 2.

(iv) $x^{100} - 1$ $x^{100} - 1$ is an expression having only non-negative integral power of x. So, it is a polynomial. Also, the highest power of x is 100, so, it is a polynomial of degree 100.

(v) $\frac{1}{\sqrt{2}} x^{2} - \sqrt{2} x + 2$ $\frac{1}{\sqrt{2}} x^{2} - \sqrt{2} x + 2$ is an expression having only non-negative integral powers of x. So, it is a polynomial. Also, the highest power of x is 2, so, it is a polynomial of degree 2.

(vi) $x^{- 2} + 2 x^{- 1} + 3$ $x^{- 2} + 2 x^{- 1} + 3$ is an expression having negative integral powers of x. So, it is not a polynomial.

(vii) Clearly, 1 is a constant polynomial of degree 0.

(viii) Clearly, $- \frac{3}{5}$ $- \frac{3}{5}$ is a constant polynomial of degree 0.

(ix) $\frac{x^{2}}{2} - \frac{2}{x^{2}} = \frac{x^{2}}{2} - 2 x^{- 2}$ $\frac{x^{2}}{2} - \frac{2}{x^{2}} = \frac{x^{2}}{2} - 2 x^{- 2}$
This is an expression having negative integral power of x i.e. −2. So, it is not a polynomial.

(x) $\sqrt[3]{2} x^{2} - 8$ $\sqrt[3]{2} x^{2} - 8$ is an expression having only non-negative integral power of x. So, it is a polynomial. Also, the highest power of x is 2, so, it is a polynomial of degree 2.

(xi) $\frac{1}{2 x^{2}} = \frac{1}{2} x^{- 2}$ $\frac{1}{2 x^{2}} = \frac{1}{2} x^{- 2}$ is an expression having negative integral power of x. So, it is not a polynomial.

(xii) $\frac{1}{\sqrt{5}} x^{\frac{1}{2}} + 1$ $\frac{1}{\sqrt{5}} x^{\frac{1}{2}} + 1$
In this expression, the power of x is $\frac{1}{2}$ $\frac{1}{2}$ which is a fraction. Since it is an expression having fractional power of x, so, it is not a polynomial.

(xiii) $\frac{3}{5} x^{2} - \frac{7}{3} x + 9$ $\frac{3}{5} x^{2} - \frac{7}{3} x + 9$ is an expression having only non-negative integral powers of x. So, it is a polynomial. Also, the highest power of x is 2, so, it is a polynomial of degree 2.

(xiv) $x^{4} - x^{\frac{3}{2}} + x - 3$ $x^{4} - x^{\frac{3}{2}} + x - 3$
In this expression, one of the powers of x is $\frac{3}{2}$ $\frac{3}{2}$ which is a fraction. Since it is an expression having fractional power of x, so, it is not a polynomial.

(xv) $2 x^{3} + 3 x^{2} + \sqrt{x} - 1 = 2 x^{3} + 3 x^{2} + x^{\frac{1}{2}} - 1$ $2 x^{3} + 3 x^{2} + \sqrt{x} - 1 = 2 x^{3} + 3 x^{2} + x^{\frac{1}{2}} - 1$
In this expression, one of the powers of x is $\frac{1}{2}$ $\frac{1}{2}$ which is a fraction. Since it is an expression having fractional power of x, so, it is not a polynomial.

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Question 2:

Identify constant, linear, quadratic, cubic and quartic polynomials from the following.
(i) –7 + x
(ii) 6y
(iii) –z³
(iv) 1 – y – y³
(v) x – x³ + x⁴
(vi) 1 + x + x²
(vii) – 6x²
(viii) – 13
(ix) – p

Answer 2:

(i) –7 + x is a polynomial with degree 1. So, it is a linear polynomial.

(ii) 6y is a polynomial with degree 1. So, it is a linear polynomial.

(iii) –z³ is a polynomial with degree 3. So, it is a cubic polynomial.

(iv) 1 – y – y³ is a polynomial with degree 3. So, it is a cubic polynomial.

(v) x – x³ + x⁴ is a polynomial with degree 4. So, it is a quartic polynomial.

(vi) 1 + x + x² is a polynomial with degree 2. So, it is a quadratic polynomial.

(vii) – 6x² is a polynomial with degree 2. So, it is a quadratic polynomial.

(viii) –13 is a polynomial with degree 0. So, it is a constant polynomial.

(ix) – p is a polynomial with degree 1. So, it is a linear polynomial.

Question 3:

Write
(i) the coefficient of $x^{3} in x + 3 x^{2} - 5 x^{3} + x^{4}$ $x^{3} in x + 3 x^{2} - 5 x^{3} + x^{4}$ .
(ii) the coefficient of x in $\sqrt{3} - 2 \sqrt{2} x + 6 x^{2}$ $\sqrt{3} - 2 \sqrt{2} x + 6 x^{2}$ .
(iii) the coefficient of x² in 2x – 3 + x³.
(iv) the coefficient of x in $\frac{3}{8} x^{2} - \frac{2}{7} x + \frac{1}{6}$ $\frac{3}{8} x^{2} - \frac{2}{7} x + \frac{1}{6}$ .
(v) the constant term in $\frac{π}{2} x^{2} + 7 x - \frac{2}{5} π$ $\frac{π}{2} x^{2} + 7 x - \frac{2}{5} π$ .

Answer 3:

(i) The coefficient of x³ in $x + 3 x^{2} - 5 x^{3} + x^{4}$ $x + 3 x^{2} - 5 x^{3} + x^{4}$ is −5.

(ii) The coefficient of x in $\sqrt{3} - 2 \sqrt{2} x + 6 x^{2}$ $\sqrt{3} - 2 \sqrt{2} x + 6 x^{2}$ is $- 2 \sqrt{2}$ $- 2 \sqrt{2}$ .

(iii) 2x – 3 + x³= – 3 + 2x + 0x²+ x³
The coefficient of x² in 2x – 3 + x³ is 0.

(iv) The coefficient of x in $\frac{3}{8} x^{2} - \frac{2}{7} x + \frac{1}{6}$ $\frac{3}{8} x^{2} - \frac{2}{7} x + \frac{1}{6}$ is $- \frac{2}{7}$ $- \frac{2}{7}$ .

(v) The constant term in $\frac{π}{2} x^{2} + 7 x - \frac{2}{5} π$ $\frac{π}{2} x^{2} + 7 x - \frac{2}{5} π$ is $- \frac{2}{5} π$ $- \frac{2}{5} π$ .

Question 4:

Determine the degree of each of the following polynomials.
(i) $\frac{4 x - 5 x^{2} + 6 x^{3}}{2 x}$ $\frac{4 x - 5 x^{2} + 6 x^{3}}{2 x}$
(ii) y²(y – y³)
(iii) (3x – 2) (2x³ + 3x²)
(iv) $- \frac{1}{2} x + 3$ $- \frac{1}{2} x + 3$
(v) – 8
(vi) x^–2(x⁴+ x²)

Answer 4:

(i) $\frac{4 x - 5 x^{2} + 6 x^{3}}{2 x} = \frac{4 x}{2 x} - \frac{5 x^{2}}{2 x} + \frac{6 x^{3}}{2 x} = 2 - \frac{5}{2} x + 3 x^{2}$ $\frac{4 x - 5 x^{2} + 6 x^{3}}{2 x} = \frac{4 x}{2 x} - \frac{5 x^{2}}{2 x} + \frac{6 x^{3}}{2 x} = 2 - \frac{5}{2} x + 3 x^{2}$
Here, the highest power of x is 2. So, the degree of the polynomial is 2.

(ii) y²(y – y³) = y³ – y⁵
Here, the highest power of y is 5. So, the degree of the polynomial is 5.

(iii) (3x – 2)(2x³ + 3x²) = 6x⁴ + 9x³ – 4x³ – 6x² = 6x⁴ + 5x³ – 6x²
Here, the highest power of x is 4. So, the degree of the polynomial is 4.

(iv) $- \frac{1}{2} x + 3$ $- \frac{1}{2} x + 3$
Here, the highest power of x is 1. So, the degree of the polynomial is 1.

(v) – 8
–8 is a constant polynomial. So, the degree of the polynomial is 0.

(vi) x^–2(x⁴+ x²) = x²+ x⁰= x²+ 1
Here, the highest power of x is 2. So, the degree of the polynomial is 2.

Question 5:

(i) Give an example of a monomial of degree 5.
(ii) Give an example of a binomial of degree 8.
(iii) Give an example of a trinomial of degree 4.
(iv) Give an example of a monomial of degree 0.

Answer 5:

(i) A polynomial having one term is called a monomial. Since the degree of required monomial is 5, so the highest power of x in the monomial should be 5.
An example of a monomial of degree 5 is 2x⁵.

(ii) A polynomial having two terms is called a binomial. Since the degree of required binomial is 8, so the highest power of x in the binomial should be 8.
An example of a binomial of degree 8 is 2x⁸ − 3x.

(iii) A polynomial having three terms is called a trinomial. Since the degree of required trinomial is 4, so the highest power of x in the trinomial should be 4.
An example of a trinomial of degree 4 is 2x⁴ − 3x + 5.

(iv) A polynomial having one term is called a monomial. Since the degree of required monomial is 0, so the highest power of x in the monomial should be 0.
An example of a monomial of degree 0 is 5.

Question 6:

Rewrite each of the following polynomials in standard form.
(i) $x - 2 x^{2} + 8 + 5 x^{3}$ $x - 2 x^{2} + 8 + 5 x^{3}$
(ii) $\frac{2}{3} + 4 y^{2} - 3 y + 2 y^{3}$ $\frac{2}{3} + 4 y^{2} - 3 y + 2 y^{3}$
(iii) $6 x^{3} + 2 x - x^{5} - 3 x^{2}$ $6 x^{3} + 2 x - x^{5} - 3 x^{2}$
(iv) $2 + t - 3 t^{3} + t^{4} - t^{2}$ $2 + t - 3 t^{3} + t^{4} - t^{2}$

Answer 6:

A polynomial written either in ascending or descending powers of a variable is called the standard form of a polynomial.

(i) $8 + x - 2 x^{2} + 5 x^{3}$ $8 + x - 2 x^{2} + 5 x^{3}$ is a polynomial in standard form as the powers of x are in ascending order.
(ii) $\frac{2}{3} - 3 y + 4 y^{2} + 2 y^{3}$ $\frac{2}{3} - 3 y + 4 y^{2} + 2 y^{3}$ is a polynomial in standard form as the powers of y are in ascending order.
(iii) $2 x - 3 x^{2} + 6 x^{3} - x^{5}$ $2 x - 3 x^{2} + 6 x^{3} - x^{5}$ is a polynomial in standard form as the powers of x are in ascending order.
(iv) $2 + t - t^{2} - 3 t^{3} + t^{4}$ $2 + t - t^{2} - 3 t^{3} + t^{4}$ is a polynomial in standard form as the powers of t are in ascending order.

RS AGGARWAL CLASS 9 Chapter 2 POLYNOMIALS EXERCISE 2A