The topics and subtopics covered in class 9 polynomials chapter 2 include:
- Polynomials in One Variable
- Zeros of Polynomials
- Remainder Theorem
- Factorization of Polynomials
- Algebraic Identities
Polynomials are expressions with one or more terms with a non-zero coefficient. A polynomial can have more than one term. In the polynomial, each expression in it is called a term. Suppose x2 + 5x + 2 is polynomial, then the expressions x2, 5x, and 2 are the terms of the polynomial. Each term of the polynomial has a coefficient. For example, if 2x + 1 is the polynomial, then the coefficient of x is 2.
The real numbers can also be expressed as polynomials. Like 3, 6, 7, are also polynomials without any variables. These are called constant polynomials. The constant polynomial 0 is called zero polynomial. The exponent of the polynomial should be a whole number. For example, x-2 + 5x + 2, cannot be considered as a polynomial, since the exponent of x is -2, which is not a whole number.
The highest power of the polynomial is called the degree of the polynomial. For example, in x3 + y3 + 3xy(x + y), the degree of the polynomial is 3. For a non zero constant polynomial, the degree is zero. Apart from these, there are other types of polynomials such as:
- Linear polynomial – of degree one
- Quadratic Polynomial- of degree two
- Cubic Polynomial – of degree three
This topic has been widely discussed in class 9 and class 10.
Example of polynomials are:
- x + y
- 7a + b + 8
- w + x + y + z
- x2 + x + 1
|Quadratic Equation||Algebraic Identities|
|Quadratic Formula & Quadratic Polynomial||Degree Of A Polynomial|
Polynomials in One Variable
Polynomials in one variable are the expressions which consist of only one type of variable in the entire expression.
Example of polynomials in one variable:
- 2x2 + 5x + 15
Some important points in Polynomials Class 9 Chapter 2 are given below:
- An algebraic expression p(x) = a0xn + a1xn-1 + a2xn-2 + … an is a polynomial where a0, a1, ………. an are real numbers and n is non-negative integer.
- A term is either a variable or a single number or it can be a combination of variable and numbers.
- The degree of the polynomial is the highest power of the variable in a polynomial.
- A polynomial of degree 1 is called as a linear polynomial.
- A polynomial of degree 2 is called a quadratic polynomial.
- A polynomial of degree 3 is called a cubic polynomial.
- A polynomial of 1 term is called a monomial.
- A polynomial of 2 terms is called binomial.
- A polynomial of 3 terms is called a trinomial.
- A real number ‘a’ is a zero of a polynomial p(x) if p(a) = 0, where a is also known as root of the equation p(x) = 0.
- A linear polynomial in one variable has a unique zero, a polynomial of a non-zero constant has no zero, and each real number is a zero of the zero polynomial.
- Remainder Theorem: If p(x) is any polynomial having degree greater than or equal to 1 and if it is divided by the linear polynomial x – a, then the remainder is p(a).
- Factor Theorem : x – c is a factor of the polynomial p(x), if p(c) = 0. Also, if x – c is a factor of p(x), then p(c) = 0.
- The degree of the zero polynomial is not defined.
- (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
- (x + y)3 = x3 + y3 + 3xy(x + y)
- (x – y)3 = x3 – y3 – 3xy(x – y)
- x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx)
Polynomials Class 9 Examples (Class 9th Maths Polynomial)
Write the coefficients of x in each of the following:
- 3x + 1
- 23x2 – 5x + 1
In 3x + 1, the coefficient of x is 3.
In 23x2 – 5x + 1, the coefficient of x is -5.
What are the degrees of following polynomials?
- 3a2 + a – 1
- 32x3 + x – 1
- 3a2 + a – 1 : The degree is 2
- 32x3 + x – 1 : The degree is 3
Polynomials Class 9 Important Questions(Class 9th Maths Polynomial)
- Find value of polynomial 2x2 + 5x + 1 at x = 3.
- Check whether x = -1/6 is zero of the polynomial p(a) = 6a + 1.
- Divide 3a2 + x – 1 by a + 1.
- Find value of k, if (a – 1) is factor of p(a) = ka2 – 3a + k.
- Factorise each of the following:
- 4x2 + 9y2 + 16z2 + 12xy – 24yx – 16xz
- 2x2 + y2 + 8z2 – 2√2xy + 4√2yz – 8xz