RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9 PDF Download

RS Aggarwal Solutions: Exercise 2A - Polynomials

Q.1. Which of the following expressions are polynomials? In case of a polynomial, write its degree.
(i) x⁵ − 2x³ + x + √3
(ii) y³ + √3y
(iii)
(iv) x¹⁰⁰ − 1
(v)
(vi) x⁻² + 2x⁻¹ + 3
(vii) 1
(viii)
(ix)
(x)
(xi)
(xii)
(xiii)

(xiv)
(xv)  2x³ + 3x² + √x − 1

Ans.
(i) x⁵ − 2x³ + x +√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) y³ + √3y 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) 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) x¹⁰⁰ − 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) 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 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,  is a constant polynomial of degree 0.

(ix)
This is an expression having negative integral power of x i.e. −2. So, it is not a polynomial.

(x) 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)is an expression having negative integral power of x. So, it is not a polynomial.

(xii)
In this expression, the power of x is 1/2 which is a fraction. Since it is an expression having fractional power of x, so, it is not a polynomial.

(xiii) 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)
In this expression, one of the powers of x is 3/2 which is a fraction. Since it is an expression having fractional power of x, so, it is not a polynomial.

(xv)
In this expression, one of the powers of x is 1/2 which is a fraction. Since it is an expression having fractional power of x, so, it is not a polynomial.

Q.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
Ans.
(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.

Q.3. Write
(i) the coefficient of x³ in x+3x² − 5x³ + x⁴.
(ii) the coefficient of x in √3−2√2x + 6x².
(iii) the coefficient of x² in 2x – 3 + x³.
(iv) the coefficient of x in
(v) the constant term in
Ans.
(i) The coefficient of x3 in x+3x²−5x³ + x⁴ is − 5.
(ii) The coefficient of x in √3 − 2√2x + 6x² is −2√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
(v) The constant term in 

Q.4. Determine the degree of each of the following polynomials.
(i)
(ii) y²(y – y³)
(iii) (3x – 2) (2x³ + 3x²)

(iv)
(v) – 8
(vi) x^–2(x⁴ + x²)
Ans.
(i)
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) 
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

Q.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.
Ans.
(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.

Q.6. Rewrite each of the following polynomials in standard form.
(i) x−2x² + 8 + 5x³

(ii) 2/3 + 4y² − 3y + 2y³
(iii) 6x³ + 2x − x⁵ − 3x²
(iv) 2 + t − 3t³ + t⁴ − t²
Ans.
A polynomial written either in ascending or descending powers of a variable is called the standard form of a polynomial.

(i) 8+x−2x²+5x³ is a polynomial in standard form as the powers of x are in ascending order.
(ii) 2/3 − 3y + 4y²+2y³ is a polynomial in standard form as the powers of y are in ascending order.
(iii) 2x−3x² + 6x³ − x⁵ is a polynomial in standard form as the powers of x are in ascending order.
(iv) 2 + t − t² − 3t³+ t⁴ is a polynomial in standard form as the powers of t are in ascending order.