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^5-2x^3+x+\sqrt{3}$$x^5-2x^3+x+\sqrt{3}$
(ii) $y^3+\sqrt{3}y$$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$$\frac{1}{\sqrt{2}}{x}^2-\sqrt{2}x+2$
(vi) $x^{-2}+2x^{-1}+3$$x^{-2}+2x^{-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]{32}{x}^2-8$$\sqrt[3]{2}{x}^2-8$
(xi) $\frac{1}{2x^2}$$\frac{1}{2x^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$$x^4-x^{\frac{3}{2}}+x-3$
(xv) $2x^3+3x^2+\sqrt{x}-1$$2x^3+3x^2+\sqrt{x}-1$

Answer 1:

(i) $x^5-2x^3+x+\sqrt{3}$$x^5-2x^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}+2x^{-1}+3$$x^{-2}+2x^{-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}-2x^{-2}$$\frac{x^2}{2}-\frac{2}{x^2}=\frac{x^2}{2}-2x^{-2}$ 
This is an expression having negative integral power of x i.e. −2. So, it is not a polynomial.

(x) $\sqrt[3]{32}{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}{2x^2}=\frac{1}{2}{x}^{-2}$$\frac{1}{2x^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) $2x^3+3x^2+\sqrt{x}-1=2x^3+3x^2+x^{\frac{1}{2}}-1$$2x^3+3x^2+\sqrt{x}-1=2x^3+3x^2+x^{\frac{1}{2}}-1$
In this expression, one of the powers of x is $\frac{1}{2}$$\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\mathrm{in}x+3x^2-5x^3+x^4$$x^3 \mathrm{in} x+3x^2-5x^3+x^4$.
(ii) the coefficient of x in $\sqrt{3}-2\sqrt{2}x+6x^2$$\sqrt{3}-2\sqrt{2}x+6x^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}$$\frac{3}{8}{x}^2-\frac{2}{7}x+\frac{1}{6}$.
(v) the constant term in $\frac{\pi }{2}{x}^2+7x-\frac{2}{5}\pi$$\frac{\pi }{2}{x}^2+7x-\frac{2}{5}\pi$.

Answer 3:

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

(ii) The coefficient of x in $\sqrt{3}-2\sqrt{2}x+6x^2$$\sqrt{3}-2\sqrt{2}x+6x^2$ is $-2\sqrt{2}$$-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{\pi }{2}{x}^2+7x-\frac{2}{5}\pi$$\frac{\mathrm{\pi }}{2}{x}^2+7x-\frac{2}{5}\mathrm{\pi }$ is $-\frac{2}{5}\pi$$-\frac{2}{5}\mathrm{\pi }$.

Question 4:

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

Answer 4:

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