Hi students, Welcome to AMBiPi (Amans Maths Blogs). In this article, you will get Polynomials CBSE NCERT Notes Class 10 Maths Chapter 2 PDF. You can download this PDF and save it in your mobile device or laptop etc.
Polynomials CBSE NCERT Notes Class 10 Maths Chapter 1 PDF
Polynomials
An algebraic expression p(x) of the form of p(x) = anxn + an-1xn-1 + an-2xn-2 + … + a2x2 + a1x + a0, where a0, a1, a2, …, an are real numbers and 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) = ax2 + bx + c.
The standard form of cubic polynomial is p(x) = ax3 + bx2 + 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) = x2 – 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) = ax2 + bx + c, is a parabola.
The parabola can be upward or downward, it depends on the sign of ‘a‘ (coefficient of x2).
If the sign of ‘a‘ (coefficient of x2) in the quadratic polynomial y = ax2 + bx + c is positive, then the graph of the quadratic polynomial is an upward parabola.
If the sign of ‘a‘ (coefficient of x2) in the quadratic polynomial y = ax2 + bx + c is negative, then the graph of the quadratic polynomial is an downward parabola.
If the graph of the quadratic polynomial p(x) = ax2 + bx + c intersects x-axis, then D > 0.
If the graph of the quadratic polynomial p(x) = ax2 + bx + c touches x-axis, then D = 0.
If the graph of the quadratic polynomial p(x) = ax2 + bx + c does NOT intersect x-axis, then D < 0.
For the quadratic polynomial, the zeroes of a quadratic polynomial ax2 + bx + c, a ≠ 0, are the x-coordinates of the points where the parabola representing y = ax2 + 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) = ax2 + 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) = ax3 + bx2 + 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) = x2 – 2x – 8 and verify the relationship between the zeroes and the coefficients.
p(x) = x2 – 2x – 8 = 0
⇒ x2 – 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[x2 – (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[x3 – (α + β + γ)x2 + (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[x2 – (α + β)x + αβ] = k[x2 – 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 3x2 – x3 – 3x + 5 by x – 1 – x2, and verify the division algorithm.
The division of – x3 + 3x2 – 3x + 5 by – x2 + x – 1 is as below.
Now, Divisor × Quotient + Remainder
= (–x2 + x – 1)(x – 2) + 3
= –x3 + x2 – x + 2x2 – 2x + 2 + 3
= –x3 + 3x2 – 3x + 5
= Dividend
Class 10 Maths Chapter 2 Examples 6:
Find all the zeroes of 2x4 – 3x3 – 3x2 + 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) = x2 – 2. Now, we divide the given polynomial by x2 – 2.
2x4 – 3x3 – 3x2 + 6x – 2 = (x2 – 2)(2x2 – 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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