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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) x− 2x+ x + √3
(ii) y+ √3y
(iii)RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
(iv) x100 − 1
(v)RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
(vi) x−2 + 2x−1 + 3
(vii) 1
(viii)RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
(ix)RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
(x)RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
(xi)RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
(xii)RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
(xiii)RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9

(xiv)RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
(xv)  2x+ 3x+ √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) y+ √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) RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9is 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) RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 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.


(vix−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, RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9 is a constant polynomial of degree 0.


(ix) RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
This is an expression having negative integral power of x i.e. −2. So, it is not a polynomial.


(x)RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9 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)RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9is an expression having negative integral power of x. So, it is not a polynomial.


(xii) RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
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) RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9is 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) RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
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)RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
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 + xis 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+3x− 5x+ 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 RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
(v) the constant term inRS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
Ans.
(i) The coefficient of x3 in x+3x2−5x+ x4 is − 5.
(ii) The coefficient of x in √3 − 2√2x + 6x2 is −2√2.

(iii) 2x – 3 + x3 = – 3 + 2x + 0x+ x3

The coefficient of xin 2x – 3 + xis 0.
(iv) The coefficient of x in RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
(v) The constant term in RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9

Q.4. Determine the degree of each of the following polynomials.
(i) RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
(ii) y2(y – y3)
(iii) (3x – 2) (2x+ 3x2)

(iv)RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
(v) – 8
(vi) x–2(x4 + x2)
Ans.
(i) RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
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) = 6x+ 9x3 – 4x3 – 6x2 = 6x4 + 5x3 – 6x2
Here, the highest power of x is 4. So, the degree of the polynomial is 4.
(iv) RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9
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−2x+ 8 + 5x3

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

The document RS Aggarwal Solutions: Polynomials- 1 | Mathematics (Maths) Class 9 is a part of the Class 9 Course Mathematics (Maths) Class 9.
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FAQs on RS Aggarwal Solutions: Polynomials- 1 - Mathematics (Maths) Class 9

1. What is a polynomial and how is it structured?
Ans. A polynomial is a mathematical expression that consists of variables raised to whole number powers, combined using addition, subtraction, and multiplication. The general form of a polynomial in one variable \(x\) is given by \(a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\), where \(a_n, a_{n-1}, ..., a_0\) are constants (coefficients), and \(n\) is a non-negative integer representing the degree of the polynomial.
2. How do you identify the degree of a polynomial?
Ans. The degree of a polynomial is identified by the highest power of the variable in the expression. For example, in the polynomial \(3x^4 + 2x^3 - x + 7\), the highest power of \(x\) is 4, so the degree of this polynomial is 4.
3. Can you explain the difference between a monomial, binomial, and trinomial?
Ans. A monomial is a polynomial with only one term, such as \(5x^2\). A binomial consists of two terms, like \(3x + 4\). A trinomial has three terms, for example, \(x^2 + 2x + 1\). The key difference lies in the number of terms present in each type of polynomial.
4. What are the common operations that can be performed on polynomials?
Ans. Common operations on polynomials include addition, subtraction, multiplication, and division. When adding or subtracting polynomials, like terms are combined. For multiplication, the distributive property is used, and for division, polynomial long division or synthetic division may be applied.
5. How can we factor polynomials and why is it important?
Ans. Factoring polynomials involves rewriting a polynomial as a product of its factors. This is achieved by identifying common factors, using techniques like grouping, or applying special product formulas. Factoring is important because it simplifies polynomial expressions, aids in solving polynomial equations, and helps in graphing functions.
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