CBSE Class 10 Mathematics Polynomials Worksheet Set 01

Question. Give examples of polynomials f(x), g(x) and r(x) which justify the division algorithm f(x) = g(x) q(x)+r(x) and (i) deg r(x) = 0 (ii) deg f(x) =deg g(x) = 2 (iii) deg q(x) = deg r(x) = 1
Answer: f(x) = g(x)Xq(x) + r(x), deg r(x) = 0
x² – 1 = (x+1)(x-1) + 0

Q.- Find which of the following algebraic expression is a polynomial.

(i) 3x² – 5x      (ii) x + 1/x

Q.- Find the zeroes of the quadratic polynomial 4x² – 9 and verify the relation between the zeroes and its coefficients.

 

Sol. We have,

4x² – 9 = (2x)² – 3² = (2x – 3) (2x + 3)

So, the value of 4x² – 9 is 0, when

2x – 3 = 0 or 2x + 3 = 0

i.e., when x =3/2

or x = – 3/2

Therefore, the zeroes of 4x² – 9 are  3/2 & – 3/2

 

Sum of the zeroes

SECTION A: (1 MARK)

Question. Form a quadratic polynomial whose zeroes are \(\frac{2}{3}\) and \(-\frac{1}{3}\).
Answer: \(9x^2 - 3x - 2\)

Question. If -1 is a zero of the polynomial \(f(x) = x^2 - 7x - 8\), then find the other zero.
Answer: \(8\)

Question. If \(\alpha\) and \(\beta\) are the zeroes of the polynomial \(2x^2 + 5x + 1\), then what is the value of \(\alpha + \beta + \alpha\beta\)?
Answer: \(-2\)

Question. If the sum of the zeroes of the polynomial \(P(x) = 3x^2 + (2k + 1)x - k + 5\) is equal to the product of the zeroes, then, find the value of k.
Answer: \(k = -6\)

Question. The graph of the polynomial \(f(x) = 2x - 5\) is a straight line. At which point does the graph intersect the x-axis?
Answer: \(\left(\frac{5}{2}, 0\right)\)

SECTION B: (2 MARKS)

Question. For what value of k, (-4) is a zero of the polynomial \(x^2 - x - (2k + 2)\)?
Answer: \(k = 9\)

Question. If m and n are the zeroes of the polynomial \(3x^2 + 11x - 4\), find the value of \(\frac{m}{n} + \frac{n}{m}\).
Answer: \(-\frac{145}{12}\)

Question. If the zeroes of the polynomial \(x^2 + px + q\) are double in value to the zeroes of the polynomial \(2x^2 - 5x - 3\), find the values of p and q.
Answer: \(p = -5, q = -6\)

Question. Form a quadratic polynomial whose one zero is \(3 + \sqrt{2}\) and the sum of zeroes is 6.
Answer: \(x^2 - 6x + 7\)

Question. If \(ax^2 - 7x + c\) has 14 as the sum of the zeroes and also as product of the zeroes, find the value of a and c.
Answer: \(a = \frac{1}{2}, c = 7\)

SECTION C: (3 MARKS)

Question. Find the zeroes of the following polynomials by factorization method and verify the relations between the zeroes and the coefficients of the polynomial.
(i) \(2x^2 - (1 + 2\sqrt{2})x + \sqrt{2}\)
(ii) \(y^2 + \frac{3\sqrt{5}}{2}y - 5\)
Answer:
(i) \(\frac{1}{2}, \sqrt{2}\)
(ii) \(-2\sqrt{5}, \frac{\sqrt{5}}{2}\)

Question. Find the value of a and b so that \(8x^4 + 14x^3 - 2x^2 + ax + b\) is exactly divisible by \(4x^2 + 3x - 2\).
Answer: \(a = -7, b = 2\)

Question. If p and q are the zeroes of the polynomial \(6y^2 - 7y + 2\), find a quadratic polynomial whose zeroes are \(\frac{1}{p}\) and \(\frac{1}{q}\).
Answer: \(\frac{1}{2}(2y^2 - 7y + 6)\)

Question. On dividing a polynomial \(3x^3 + 4x^2 + 5x - 13\) by a polynomial g(x), the quotient and the remainder were (3x + 10) and (16x – 43) respectively. Find g(x).
Answer: \(x^2 - 2x + 3\)

Question. If one zero of a polynomial \(3x^2 - 8x + 2k + 1\) is seven times the other, find the value of k.
Answer: \(k = \frac{2}{3}\)

SECTION D: (4 MARKS)

Question. Find the other zeroes of the polynomial \(P(x) = 2x^4 + 7x^3 - 19x^2 - 14x + 30\), if two of its zeroes are \(\frac{3}{2}\) and \(-5\).
Answer: \(-\sqrt{2}, \sqrt{2}\)

Question. Given \(\sqrt{2}\) is a zero of the cubic polynomial \(6x^3 + \sqrt{2}x^2 - 10x - 4\sqrt{2}\), find the other two zeroes.
Answer: \(-\frac{\sqrt{2}}{2}, -\frac{2\sqrt{2}}{3}\)

Question. If the polynomial \(x^4 - 6x^3 + 16x^2 - 25x + 10\) is divided by another polynomial \(x^2 - 2x + k\), the remainder comes out to be x + a, find the values of k and a.
Answer: \(k = 5, a = -5\)

Question. If the remainder on division of \(x^3 + 2x^2 + kx + 3\) by \(x - 3\) is 21, find the quotient and the value of k. Hence find the zeroes of the cubic polynomial \(x^3 + 2x^2 + kx - 18\).
Answer: \(k = -9\), Quotient \(= x^2 + 5x + 6\), Zeroes: \(3, -2, -3\)

Question. If \(\alpha\) and \(\beta\) are the zeroes of the polynomial \(p(x) = 2x^2 + 5x + k\) satisfying the relation \(\alpha^2 + \beta^2 + \alpha\beta = \frac{21}{4}\), then find the value of k.
Answer: \(k = 2\)