Polynomials Class 10 Maths NCERT Solutions
NCERT Solutions for Class 10 Maths Chapter 2 Polynomials are part of NCERT Solutions for Class 10 Maths. Here we have given Maths NCERT Solutions Class 10 Chapter 2 Polynomials.
Class 10 Maths NCERT Solutions Chapter 2 Polynomials Ex 2.1
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.
Class 10 Maths NCERT Solutions Chapter 2 Polynomials Ex 2.2
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and their coefficients:
(i) x2 – 2x – 8
(ii) 4s2 – 4s + 1
(iii) 6x2 – 3 – 7x
(iv) 4u2 + 8u
(v) t2 – 15
(vi) 3x2 – x – 4
Find a quadratic polynomial each with the given numbers as the sum and product of zeroes respectively:
Class 10 Maths NCERT Solutions Chapter 2 Polynomials Ex 2.3
Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:
(i) p(x) = x3 – 3x2 + 5x – 3, g(x) = x2 – 2
(ii) p(x) = x4 – 3x2 + 4x + 5, g(x) = x2 + 1 – x
(iii) p(x) = x4– 5x + 6, g(x) = 2 – x2
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.
(i) t2 – 3, 2t4 + 3t3 – 2t2– 9t – 12
(ii) x2 + 3x + 1, 3x4 + 5x3 – 7x2 + 2x + 2
(iii) x2 + 3x + 1, x5 – 4x3 + x2 + 3x + 1
Obtain all other zeroes of 3x4 + 6x3 – 2x2 – 10x – 5, if two of its zeroes are and and –
On dividing x3 – 3x2 + x + 2bya polynomial g(x), the quotient and remainder were x – 2 and -2x + 4 respectively. Find g(x).
Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and:
(i) deg p(x) = deg q(x)
(ii) deg q(x) = deg r(x)
(iii) deg r(x) = 0
Class 10 Maths NCERT Solutions Chapter 2 Polynomials Ex 2.4
Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also, verify the relationship between the zeroes and the coefficients in each case:
(i) 2x3 + x2 – 5x + 2; , 1, -2
(ii) x3 – 4x2 + 5x – 2; 2, 1, 1
Find a cubic polynomial with the sum, some of the product of its zeroes taken two at a time, and the product of its zeroes as 2, -7, -14 respectively.
If the zeroes of the polynomial x3 – 3x2 + x + 1 are a-b, a, a + b, find a and b.
If two zeroes of the polynomial x4 – 6x3 – 26x2 + 138x – 35 are 2 ± √3, find other zeroes.
If the polynomial x4 – 6x3 + 16x2 – 25x + 10 is divided by another polynomial x2 – 2x + k, the remainder comes out to be x + a, find k and a.
Class 10 Maths Polynomials Mind Map
An algebraic expression f(x) of the form
f(x) = a0 + a1x + a2x2 +…. + anxn,
where a0, a1, ……., an are real numbers and all indices of variables are non-negative integers is called polynomial in variable x.
(i) The highest power of x is called degree of the polynomial.
(ii) a0, a1x,….,anxn are terms of the polynomial.
(iii) a0, a1 ,….an are co-efficients of the polynomial.
Standard Forms of Linear, Quadratic and Cubic Polynomials
(i) Linear Polynomial:
ax + b, where a, b are real numbers and a ≠ 0.
(ii) Quadratic Polynomial:
ax2 + bx + c, where a, b, c are real numbers & a ≠ 0.
(iii) Cubic Polynomials:
ax3 + bx2 + cx + d, where a, b, c, d are real numbers and a ≠ 0.
Value of a Polynomial
The value of a polynomial f(x) at x = a is obtained by substituting x = a in the given polynomial and is denoted by/(a).
Zero(es)/Root(s) of Polynomial
x = r is a zero of a polynomial p(x) if p(r) = 0.
Geometrical Meaning of Zeroes of a Polynomial
Zero(es) of a polynomial is/are the x-coordinate of the point(s) where graph y = fix) intersects the x-axis.
(i) Linear polynomial: Graph of linear polynomial is a straight line and has exactly one zero.
(ii) Quadratic polynomial: Graph of quadratic polynomial is always a parabola and this polynomial can have atmost two zeroes.
(iii) Cubic polynomial: Cubic polynomial can have atmost three zeroes.
Cases of Quadratic Polynomial
Case-I : If a quadratic polynomial P(x) = ax2 + bx + c has two zeroes, then its graph will intersect the x-axis at two distinct points A and B as shown in the figure.
Case-II: If a quadratic polynomial P(x) = ax2 + bx + c has only one zero, then its graph will touch the x-axis at only one point A as shown in the figure.
Case-III : If a quadratic polynomial P(x) = ax2 + bx + c has no zero, then its graph will not intersect /touch the x-axis at any point as shown in the figure.
Relationship between Zeroes and Coefficients of a Polynomial
(i) Zero of a linear polynomial ax + b is x =
(ii) If α and β are the zeroes of the quadratic polynomial ax2 + bx + c, then
α + β =
(iii) If α, β and γ are zeroes of the cubic polynomial ax3 + bx2 + cx + d then
If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x)
where either r(x) = 0 or degree of r(x) < degree of g(x)
NCERT Solutions for Class 10 Maths
- Chapter 1 Real Numbers
- Chapter 2 Polynomials
- Chapter 3 Pair of Linear Equations in Two Variables
- Chapter 4 Quadratic Equations
- Chapter 5 Arithmetic Progressions
- Chapter 6 Triangles
- Chapter 7 Coordinate Geometry
- Chapter 8 Introduction to Trigonometry
- Chapter 9 Some Applications of Trigonometry
- Chapter 10 Circles
- Chapter 11 Constructions
- Chapter 12 Areas Related to Circles
- Chapter 13 Surface Areas and Volumes
- Chapter 14 Statistics
- Chapter 15 Probability