Class 10 Maths Polynomials Notes

Introduction

In Class 10, polynomials are one of the essential algebraic concepts. A polynomial is a mathematical expression involving a sum of powers of a variable, each multiplied by a coefficient. Understanding polynomials is crucial as they are used in various fields like geometry, physics, economics, and even in solving real-world problems. In this blog post, we will explore the key concepts from the Polynomials chapter of Class 10 Maths, including the geometrical meaning of zeroes, the relationship between zeroes and coefficients, and more.

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2.1 Introduction to Polynomials

A polynomial is an algebraic expression consisting of terms that are either constants or variables raised to whole-number powers. A general polynomial in one variable $x$x can be written as:$$P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$P(x)=an​xn+an−1​xn−1+⋯+a1​x+a0​

Where:

• $a_n, a_{n-1}, \dots, a_1, a_0$an​,an−1​,…,a1​,a0​ are the coefficients of the polynomial.
• $n$n is a non-negative integer, which represents the degree of the polynomial.
• $x$x is the variable.

For example:

• $4x^3 + 3x^2 – 2x + 1$4×3+3×2−2x+1 is a cubic polynomial.
• $5x^2 + 3x + 2$5×2+3x+2 is a quadratic polynomial.
• $7x + 9$7x+9 is a linear polynomial.

Polynomials are classified based on their degree:

• Degree 0: Constant polynomials (e.g., $5$5).
• Degree 1: Linear polynomials (e.g., $3x + 4$3x+4).
• Degree 2: Quadratic polynomials (e.g., $x^2 + 2x + 1$x2+2x+1).
• Degree 3: Cubic polynomials (e.g., $x^3 – 3x^2 + x – 2$x3−3×2+x−2).

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2.2 Geometrical Meaning of the Zeroes of a Polynomial

The zeroes of a polynomial are the values of $x$x for which $P(x) = 0$P(x)=0. Geometrically, the zeroes of a polynomial are the points where the graph of the polynomial intersects the x-axis.

For a linear polynomial $P(x) = ax + b$P(x)=ax+b:

The graph is a straight line. It intersects the x-axis at one point, which is the zero of the polynomial.

For a quadratic polynomial $P(x) = ax^2 + bx + c$P(x)=ax2+bx+c:

The graph is a parabola. The zeroes are the points where the parabola intersects the x-axis. Depending on the discriminant ($b^2 – 4ac$b2−4ac), the number of real zeroes can be:

• Two distinct real zeroes (if $b^2 – 4ac > 0$b2−4ac>0)
• One real zero (if $b^2 – 4ac = 0$b2−4ac=0)
• No real zeroes (if $b^2 – 4ac < 0$b2−4ac<0).

For a cubic polynomial $P(x) = ax^3 + bx^2 + cx + d$P(x)=ax3+bx2+cx+d:

The graph is a curve, and it may intersect the x-axis at one, two, or three points, depending on the nature of the polynomial and its coefficients.

In general, the number of zeroes of a polynomial is equal to its degree (counting multiplicities). For example, a cubic polynomial can have up to three zeroes.

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2.3 Relationship Between Zeroes and Coefficients of a Polynomial

The relationship between the zeroes and coefficients of a polynomial is a key concept. For a polynomial of the form:$$P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$P(x)=an​xn+an−1​xn−1+⋯+a1​x+a0​

Let the zeroes of the polynomial be $\alpha, \beta, \gamma, \dots$α,β,γ,… (for a polynomial of degree $n$n). The relationship between the zeroes and the coefficients can be summarized as follows:

For a quadratic polynomial $P(x) = ax^2 + bx + c$P(x)=ax2+bx+c:

• The sum of the zeroes $\alpha + \beta = -\frac{b}{a}$α+β=−ab​
• The product of the zeroes $\alpha \beta = \frac{c}{a}$αβ=ac​

For a cubic polynomial $P(x) = ax^3 + bx^2 + cx + d$P(x)=ax3+bx2+cx+d:

• The sum of the zeroes $\alpha + \beta + \gamma = -\frac{b}{a}$α+β+γ=−ab​
• The sum of the products of the zeroes taken two at a time $\alpha \beta + \beta \gamma + \gamma \alpha = \frac{c}{a}$αβ+βγ+γα=ac​
• The product of the zeroes $\alpha \beta \gamma = -\frac{d}{a}$αβγ=−ad​

This relationship is helpful in solving polynomial equations and understanding how the coefficients affect the roots.

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2.4 Summary

In this chapter, we have explored the fundamental concepts of polynomials, including their definitions, types, and the important relationships between zeroes and coefficients.

Key takeaways:

• A polynomial is an expression made up of terms involving powers of a variable.
• The zeroes of a polynomial are the values of $x$x where the polynomial equals zero, and they correspond to the points where the graph intersects the x-axis.
• There is a specific relationship between the zeroes of a polynomial and its coefficients, which is crucial for solving polynomial equations.

Mastering these concepts will help students solve a wide range of problems involving polynomials in both algebraic and geometrical contexts.

MCQs Based on the “Polynomials” Chapter:

1. What is the degree of the polynomial 4×3+3×2−2x+14x^3 + 3x^2 – 2x + 14×3+3×2−2x+1?

a) 1
b) 2
c) 3
d) 4

Answer: c) 3

2. Which of the following is a quadratic polynomial?

a) $x^3 + 2x^2 + 1$x3+2×2+1
b) $x^2 + 5x + 6$x2+5x+6
c) $3x + 7$3x+7
d) $2x^4 + 3x^2 + 1$2×4+3×2+1

Answer: b) $x^2 + 5x + 6$x2+5x+6

3. If the zeroes of the quadratic polynomial x2−5x+6x^2 – 5x + 6×2−5x+6 are α\alphaα and β\betaβ, what is the value of α+β\alpha + \betaα+β?

a) 6
b) 5
c) -6
d) -5

Answer: b) 5

4. For a cubic polynomial, the sum of the zeroes α,β,γ\alpha, \beta, \gammaα,β,γ is given by:

a) $-\frac{b}{a}$−ab​
b) $\frac{b}{a}$ab​
c) $\frac{a}{b}$ba​
d) $-\frac{a}{b}$−ba​

Answer: a) $-\frac{b}{a}$−ab​