Polynomial Class 10

Class 10 Maths - Polynomials: NCERT Solutions and Comprehensive GuideWelcome to our comprehensive guide on the Polynomials chapter for Class 10 Maths (NCERT). This blog post will cover all the essential concepts, formulas, and solutions to the NCERT textbook exercises. Polynomials are a fundamental topic in algebra, and a strong understanding of this chapter is crucial for your future mathematical studies.What are Polynomials?A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, and non-negative integer exponents of variables.Variables: Symbols that represent unknown values (e.g., x, y, z).Coefficients: Numerical values multiplied by the variables (e.g., in 2x2, 2 is the coefficient).Terms: Parts of the expression separated by addition or subtraction (e.g., 2x2, -3x, 5 are terms).Degree: The highest power of the variable in the polynomial.Types of Polynomials (Based on Degree):Linear Polynomial: Degree 1 (e.g., ax + b).Quadratic Polynomial: Degree 2 (e.g., ax2 + bx + c).Cubic Polynomial: Degree 3 (e.g., ax3 + bx2 + cx + d).Zeroes of a PolynomialThe zeroes of a polynomial p(x) are the values of x for which p(x) = 0. Geometrically, the zeroes of a polynomial are the x-coordinates of the points where the graph of the polynomial intersects the x-axis.Relationship Between Zeroes and CoefficientsFor a quadratic polynomial ax2 + bx + c, if α and β are the zeroes, then:Sum of zeroes: α + β = -b/aProduct of zeroes: αβ = c/aDivision Algorithm for PolynomialsIf 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 r(x) = 0 or degree of r(x) < degree of g(x).p(x) is the dividend.g(x) is the divisor.q(x) is the quotient.r(x) is the remainder.NCERT Solutions for Class 10 Maths - Polynomials

Now, let's dive into the solutions for the exercises in the NCERT textbook

.Exercise 2.1

The graphs of y = p(x) are given in the figure below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.(i) The graph intersects the x-axis at one point. Number of zeroes = 1.(ii) The graph intersects the x-axis at two points. Number of zeroes = 2.(iii)The graph intersects the x-axis at three points. Number of zeroes = 3(iv) The graph intersects the x-axis at two points. Number of zeroes = 2(v) The graph intersects the x-axis at four points. Number of zeroes = 4(vi) The graph intersects the x-axis at three points. Number of zeroes = 3

Exercise 2.2

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.(i) x2 - 2x - 8Solution: x2 - 2x - 8 = (x - 4)(x + 2). Zeroes are 4 and -2.Sum of zeroes = 4 + (-2) = 2 = -(-2)/1 = -b/aProduct of zeroes = 4 * (-2) = -8 = -8/1 = c/a(ii) 4s2 - 4s + 1Solution: 4s2 - 4s + 1 = (2s-1)2. Zero is 1/2.Sum of zeroes = 1/2 + 1/2 = 1 = -(-4)/4 = -b/aProduct of zeroes = 1/2 * 1/2 = 1/4 = 1/4 = c/a(iii) 6x2 - 3 - 7xSolution: 6x2 - 7x - 3 = (3x + 1)(2x - 3). Zeroes are -1/3 and 3/2Sum of zeroes = -1/3 + 3/2 = 7/6 = -(-7)/6 = -b/aProduct of zeroes = -1/3 * 3/2 = -1/2 = -3/6 = c/a(iv) 4u2 + 8uSolution: 4u2 + 8u = 4u(u+2). Zeroes are 0 and -2.Sum of zeroes = 0 + (-2) = -2 = -(8)/4 = -b/aProduct of zeroes = 0 * -2 = 0 = 0/4 = c/a(v) t2 - 15Solution: t2 - 15 = (t - √15)(t + √15). Zeroes are √15 and -√15.Sum of zeroes = √15 - √15 = 0 = -(0)/1 = -b/aProduct of zeroes = (√15)(-√15) = -15 = -15/1 = c/a(vi) 3x2 - x - 4Solution: 3x2 - x - 4 = (3x - 4)(x + 1). Zeroes are 4/3 and -1.Sum of zeroes = 4/3 + (-1) = 1/3 = -(-1)/3 = -b/aProduct of zeroes = 4/3 * (-1) = -4/3 = -4/3 = c/aFind a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.(i) 1/4, -1Solution: x2 - (1/4)x + (-1) = 4x2 - x - 4(ii) √2, 1/3Solution: x2 - (√2)x + 1/3 = 3x2 - 3√2x + 1(iii) 0, √5Solution : x2 - (0)x + √5 = x2 + √5(iv) 1,1Solution : x2 - (1)x + 1 = x2 - x + 1(v) -1/4, 1/4Solution : x2 - (-1/4)x + 1/4 = 4x2 + x + 1(vi) 4,1Solution : x2 - (4)x + 1 = x2 - 4x + 1

Exercise 2.3

Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder.(i) p(x) = x3 - 3x2 + 5x - 3, g(x) = x2 - 2Solution: Quotient = x - 3, Remainder = 7x - 9(ii) p(x) = x4 - 3x2 + 4x + 5, g(x) = x2 + 1 - xSolution: Quotient = x2 + x - 3, Remainder = 8(iii) p(x) = x4 - 5x + 6, g(x) = 2 - x2Solution: Quotient = -x2 - 2, Remainder = -5x + 10Check 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 - 12Solution: Yes, t2 - 3 is a factor (Remainder = 0).(ii) x2 + 3x + 1, 3x4 + 5x3 - 7x2 + 2x + 2Solution: No, x2 + 3x + 1 is not a factor (Remainder ≠ 0).(iii) x3 - 3x + 1, x5 - 4x3 + x2 + 3x + 1Solution: No, x3 - 3x + 1 is not a factor (Remainder ≠ 0).Obtain all other zeroes of 3x4 + 6x3 - 2x2 - 10x - 5, if two of its zeroes are √5/3 and -√5/3.Solution: Other zeroes are -1 and -1.On dividing x3 - 3x2 + x + 2 by a polynomial g(x), the quotient and remainder were x - 2 and -2x + 4, respectively. Find g(x).Solution: g(x) = x2 - x + 1Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and(i) degree of p(x) = degree of q(x)Example: p(x) = 2x2 - 2x + 4, g(x) = 2, q(x) = x2 - x + 2, r(x) = 0(ii) degree of q(x) = degree of r(x)Example: p(x) = x3 + x2 + x + 1, g(x) = x2 - 1, q(x) = x + 1, r(x) = 2x + 2(iii) degree of r(x) = 0Example: p(x) = x2 + 2x + 1, g(x) = x + 1, q(x) = x + 1, r(x) = 0

Exercise 2.4 

(Optional)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, 1, -2(ii) x3 - 4x2 + 5x - 2; 2, 1, 1Find a cubic polynomial with the sum, sum 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.(Note: Solutions for Exercise 2.4 are not provided here, but can be found on several online resources. If you'd like me to provide those as well, just ask!)Key TakeawaysPolynomials are algebraic expressions with specific rules.The degree of a polynomial determines its type.Zeroes of a polynomial are the values of the variable that make the polynomial equal to zero.There's a relationship between the zeroes and coefficients of a polynomial.The division algorithm helps in dividing polynomials.This blog post provides a comprehensive overview of the Polynomials chapter for Class 10 Maths. By understanding the concepts and practicing the NCERT questions, you can build a strong foundation in this important topic. If you have any doubts, feel free to ask in the comments section.

Sample Test paper 

Que  2 - If 2 and 3 are the zeros of the polynomial 3x²-2kx+2m = 0 then value of k and m are 

a) 15,9/2

b) 15/2  ,- 9

c) 15/2,. 9

d ) none

Que 3 -- A polynomial whose sum and product of zeros is 3 and 2. Find the equation of polynomial

a ) x²+3x+2

b) x²-3x-2

c) x²-3x + 2

d) none of these 

Que 4 --If 1 is the zero of the polynomial. 

3x² +5x + a find the value of a 

a) -8

b) 8

c) 7

d). - 7

Thu Jan 12, 2023