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Polynomials – Class 9 Mathematics
Polynomials are one of the most important topics in Class 9 Mathematics.
They are widely used in algebra, geometry, and higher-level mathematics.
In this blog, we will understand what polynomials are, their types, degrees,
and solve examples step by step.
1. What is a Polynomial?
An algebraic expression that contains variables, coefficients, and
non-negative integral powers of variables is called a polynomial.
The general form of a polynomial in one variable \( x \) is:
\[
a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0
\]
where \( a_0, a_1, a_2, \dots, a_n \) are real numbers and \( n \) is a non-negative integer.
2. Examples of Polynomials
- \( 3x^2 + 5x + 7 \)
- \( 4y – 9 \)
- \( 2x^3 – x \)
Not Polynomials
- \( \frac{1}{x} + 2 \) (power of \( x \) is -1)
- \( \sqrt{x} + 3 \) (power of \( x \) is \( \frac{1}{2} \))
3. Degree of a Polynomial
The degree of a polynomial is the highest power of the variable in the expression.
\[
5x^3 + 2x^2 – 7
\]Solution:
The highest power of \( x \) is 3.
Therefore, the degree of the polynomial is 3.
4. Types of Polynomials (Based on Degree)
- Constant Polynomial: Degree 0 (e.g., \( 7 \))
- Linear Polynomial: Degree 1 (e.g., \( 2x + 3 \))
- Quadratic Polynomial: Degree 2 (e.g., \( x^2 + 5x + 6 \))
- Cubic Polynomial: Degree 3 (e.g., \( x^3 – 4x + 1 \))
5. Number of Terms in a Polynomial
- Monomial: One term (e.g., \( 5x \))
- Binomial: Two terms (e.g., \( x + 2 \))
- Trinomial: Three terms (e.g., \( x^2 + x + 1 \))
6. Value of a Polynomial
The value of a polynomial is obtained by substituting a numerical value
for the variable.
\[
p(x) = 2x^2 – 3x + 1
\]
at \( x = 2 \).Solution:
\[
p(2) = 2(2)^2 – 3(2) + 1
\]
\[
= 2(4) – 6 + 1
\]
\[
= 8 – 6 + 1 = 3
\]
7. Zero of a Polynomial
A number \( k \) is called a zero of a polynomial \( p(x) \)
if \( p(k) = 0 \).
\[
p(x) = x – 1
\]Solution:
\[
p(1) = 1 – 1 = 0
\]
Since \( p(1) = 0 \), 1 is a zero of the polynomial.