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# Polynomials CBSE NCERT Notes Class 10 Maths Chapter 1 PDF

## Polynomials

An algebraic expression p(x) of the form of **p(x) = a _{n}x^{n} + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + … + a_{2}x^{2} + a_{1}x + a_{0}**, where

**a**are real numbers and

_{0}, a_{1}, a_{2}, …, a_{n}**n**is a non-negative integers.

The highest exponent **n** of the polynomial is the **degree** of the polynomial.

The degree of a constant polynomial p(x) = k is zero.

The degree of a zero polynomial p(x) = 0 is not defined.

The standard form of **linear polynomial** is **p(x) = ax + b**.

The standard form of **quadratic polynomial** is **p(x) = ****ax ^{2} + bx + c**.

The standard form of **cubic polynomial** is **p(x) = ****ax ^{3} + bx^{2} + cx + d**.

## Value of a Polynomial

If p(x) is a polynomial in x and k is any real number, then the value of the polynomial p(x) is obtained by replacing x by k in p(x), and is denoted by p(k).

### Class 10 Maths Chapter 2 Examples 1:

Find the value of p(x) = x^{2} – 3x – 4 at x = 2.

The value of given polynomial at x = 2 is p(2) = (2)^{2} – 3(2) – 4 = 4 – 6 – 4 = – 6.

## Geometrical Meaning of Zeroes of a Polynomial

A real number k is a zero of the polynomial p(x) if p(k) = 0.

For a **linear polynomial** p(x) = ax + b, a ≠ 0, the graph of y = ax + b is a straight line which intersects the x-axis at exactly one point, (-b/a, 0).

Thus, a linear polynomial p(x) = ax + b, a ≠ 0, has exactly one zero, the x-coordinate of the point where the graph of y = ax + b intersects the x-axis.

The graph of a **quadratic polynomial** **y =** **p(x) = ax ^{2} + bx + c**, is a

**parabola**.

The parabola can be upward or downward, it depends on the sign of ‘**a**‘ (coefficient of x^{2}).

If the sign of ‘**a**‘ (coefficient of x^{2}) in the quadratic polynomial y = ax^{2} + bx + c is positive, then the graph of the quadratic polynomial is an upward parabola.

If the sign of ‘**a**‘ (coefficient of x^{2}) in the quadratic polynomial y = ax^{2} + bx + c is negative, then the graph of the quadratic polynomial is an downward parabola.

If the graph of the quadratic polynomial p(x) = ax^{2} + bx + c **intersects x-axis**, then **D > 0**.

If the graph of the quadratic polynomial p(x) = ax^{2} + bx + c **touches x-axis**, then **D = 0**.

If the graph of the quadratic polynomial p(x) = ax^{2} + bx + c **does NOT intersect x-axis**, then **D < 0**.

For the quadratic polynomial, the zeroes of a quadratic polynomial ax^{2} + bx + c, a ≠ 0, are the x-coordinates of the points where the parabola representing y = ax^{2} + bx + c intersects the x-axis.

Thus, for a given polynomial p(x) of degree n, the graph of y = p(x) intersects the x-axis at atmost n points. Therefore, a polynomial p(x) of degree n has at most n zeroes.

### Class 10 Maths Chapter 2 Examples 2:

The graphs of y = p(x) are given below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

In the graph (i), the line does not intersect x-axis. Thus, the number of zeroes of p(x) is zero.

In the graph (ii), the curve intersects x-axis at one point only. Thus, the number of zeroes of p(x) is one.

In the graph (iii), the curve intersects x-axis at three points. Thus, the number of zeroes of p(x) is three.

## Relations Between Zeroes and Coefficients of Polynomials

For **quadratic polynomial** **p(x) = ax ^{2} + bx + c**, sum of the roots is

**α + β = -b/a**and product of the roots is

**αβ = c/a**.

For **cubic polynomial** **p(x) = ax ^{3} + bx^{2} + cx + d**, sum of the roots is

**α + β + γ = -b/a**and the sum of product of roots two at a time is

**αβ + βγ + γα = c/a**and the product of the roots is

**αβγ = -d/a**.

### Class 10 Maths Chapter 2 Examples 3:

Find the zeroes of the quadratic polynomial p(x) = x^{2} – 2x – 8 and verify the relationship between the zeroes and the coefficients.

p(x) = x^{2} – 2x – 8 = 0

⇒ x^{2} – 4x + 2x – 8 = 0

⇒ x(x – 4) + 2(x – 4) = 0

⇒ (x + 2)(x – 4) = 0

⇒ x = -2 or 4

α + β = (-2) + 4 = 2 = -(-2)/1 = -b/a

αβ = (-2) × 4 = -8 = -8/1 = c/a

## Formation of Polynomials

If two roots **α **and** β** are given, then the **quadratic polynomial** **p(x) = k(x – α)(x – β) = k[x ^{2} – (α + β)x + αβ]**

If three roots **α,**** β** and **γ** are given, then the **cubic polynomial** **p(x) = k(x – α)(x – β)(x – γ) = k[x ^{3} – (α + β + γ)x^{2} + (αβ + βγ + γα)x – αβγ].**

### Class 10 Maths Chapter 2 Examples 4:

Find a quadratic polynomial with the sum of the zeroes is 4 and the product of its zeroes is 1.

Given that α = 4 and β = 1.

Thus, the quadratic polynomial is p(x) = k[x^{2} – (α + β)x + αβ] = k[x^{2} – 5x + 4].

## Division Algorithm of Polynomials

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then there exist the polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).

### Class 10 Maths Chapter 2 Examples 5:

Divide 3x^{2} – x^{3} – 3x + 5 by x – 1 – x^{2}, and verify the division algorithm.

The division of – x^{3} + 3x^{2 }– 3x + 5 by – x^{2} + x – 1 is as below.

Now, Divisor × Quotient + Remainder

= (–x^{2} + x – 1)(x – 2) + 3

= –x^{3} + x^{2} – x + 2x^{2} – 2x + 2 + 3

= –x^{3} + 3x^{2} – 3x + 5

= Dividend

### Class 10 Maths Chapter 2 Examples 6:

Find all the zeroes of 2x^{4} – 3x^{3} – 3x^{2} + 6x – 2, if you know that two of its zeroes as √2 or –√2.

Given that two zeroes are √2 and −√2. Thus, one of the factors of the given polynomial is (x + √2)(x – √2) = x^{2} – 2. Now, we divide the given polynomial by x^{2} – 2.

2x^{4} – 3x^{3} – 3x^{2} + 6x – 2 = (x^{2} – 2)(2x^{2} – 3x + 1) = (x + √2)(x – √2)(2x – 1)(x – 1).

Thus, all of its zeroes are – √2, √2, 1/2, 1.

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