NCERT Exemplar Solutions
Class 10 - Mathematics

CHAPTER 2: Polynomials

Answer the following and justify:
(i) Can \(x^2-1\) be the quotient on division of \(x^6+2x^3+x-1\) by a polynomial in \(x\) of degree 5?
(ii) What will the quotient and remainder be on division of \(ax^2+bx+c\) by \(px^3+qx^2+rx+s\), \(p\ne0\)?
(iii) If on division of a polynomial \(p(x)\) by a polynomial \(g(x)\), the quotient is zero, what is the relation between \(\deg p\) and \(\deg g\)?
(iv) If on division of a non-zero polynomial \(p(x)\) by a polynomial \(g(x)\), the remainder is zero, what is the relation between \(\deg p\) and \(\deg g\)?
(v) Can the quadratic polynomial \(x^2+kx+k\) have equal zeroes for some odd integer \(k>1\)?

Are the following statements True/False? Justify:
(i) If the zeroes of a quadratic polynomial \(ax^2+bx+c\) are both positive, then \(a,b,c\) all have the same sign.
(ii) If the graph of a polynomial intersects the x-axis at only one point, it cannot be a quadratic polynomial.
(iii) If the graph of a polynomial intersects the x-axis at exactly two points, it need not be a quadratic polynomial.
(iv) If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
(v) If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term have the same sign.
(vi) If all three zeroes of a cubic polynomial \(x^3+ax^2-bx+c\) are positive, then at least one of \(a,b,c\) is non-negative.
(vii) The only value of \(k\) for which the quadratic polynomial \(kx^2+x+k\) has equal zeroes is \(\dfrac{1}{2}\).

Find the zeroes of \(4x^2-3x-1\) by factorisation and verify relations.

Find the zeroes of \(3x^2+4x-4\) and verify relations.

Find the zeroes of \(5t^2+12t+7\) and verify relations.

Find the zeroes of \(t^3-2t^2-15t\) and verify relations.

Find the zeroes of \(2x^2+\dfrac{7}{2}x+\dfrac{3}{4}\) and verify relations.

Find the zeroes of \(4x^2+5\sqrt{2}x-3\) and verify relations.

Find the zeroes of \(2s^2-(1+2\sqrt{2})s+\sqrt{2}\) and verify relations.

Find the zeroes of \(v^2+4\sqrt{3}v-15\) and verify relations.

Find the zeroes of \(y^2+\dfrac{\sqrt{3}}{2}y-5\) and verify relations.

Find the zeroes of \(7y^2-\dfrac{11}{3}y-23\) and verify relations.

For each of the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.

(i). \(-\dfrac{8}{3}\), \(\dfrac{4}{3}\)

(ii). \(\dfrac{21}{8}\), \(\dfrac{5}{16}\)

(iii). \(-2\sqrt{3}\), \(-9\)

(iv). \(-\dfrac{3}{2\sqrt{5}}\), \(-\dfrac{1}{2}\)

Given that the zeroes of \(x^3-6x^2+3x+10\) are of the form \(a,\, a+b,\, a+2b\), find \(a\), \(b\) and the zeroes.

Given that \(\sqrt{2}\) is a zero of \(6x^3 + \sqrt{2}x^2 - 10x - 4\sqrt{2}\), find the other two zeroes.

Find \(k\) so that \(x^2 + 2x + k\) is a factor of \(2x^4 + x^3 - 14x^2 + 5x + 6\). Also find all zeroes of the two polynomials.

Given that \(x - 5\) is a factor of \(x^3 - 3\sqrt{2}\,x^2 + 13x - 3\sqrt{5}\), find all zeroes.

For which values of \(a\) and \(b\), are the zeroes of \(q(x)=x^3+2x^2+a\) also the zeroes of \(p(x)=x^5-x^4-4x^3+3x^2+3x+b\)? Which zeroes of \(p(x)\) are not zeroes of \(q(x)\)?