NCERT Class X Mathematics Examplar Chapter 2

Chapter 2: Polynomials

(A) Main Concepts and Results

Geometrical meaning of zeroes of a polynomial: The zeroes of a polynomial are precisely the x-coordinates of the points where the graph of $y=p(x)$ intersects the x-axis.

Relation between the zeroes and coefficients of a polynomial:

• If $\alpha$ and $\beta$ are the zeroes of a quadratic polynomial $ax^{2}+bx+c$, then $\alpha+\beta=-\frac{b}{a}$ and $\alpha\beta=\frac{c}{a}$.
• If $\alpha$, $\beta$ and $\gamma$ are the zeroes of a cubic polynomial $ax^{3}+bx^{2}+cx+d$, then:
  • $\alpha+\beta+\gamma=-\frac{b}{a}$
  • $\alpha\beta+\beta\gamma+\gamma\alpha=\frac{c}{a}$
  • $\alpha\beta\gamma=\frac{-d}{a}$

The division algorithm: Given any polynomial $p(x)$ and any non-zero polynomial $g(x)$, there are polynomials $q(x)$ and $r(x)$ such that $p(x)=g(x)q(x)+r(x)$ where $r(x)=0$ or degree $r(x) <$ degree $g(x)$.

(B) Multiple Choice Questions

Choose the correct answer from the given four options:

Sample Question 1: If one zero of the quadratic polynomial $x^{2}+3x+k$ is 2, then the value of k is

1. 10
2. -10
3. 5
4. -5

Solution: Answer (B)

Sample Question 2: Given that two of the zeroes of the cubic polynomial $ax^{3}+bx^{2}+cx+d$ are 0, the third zero is

1. $\frac{-b}{a}$
2. $\frac{b}{a}$
3. $\frac{c}{a}$
4. $-\frac{d}{a}$

Solution: Answer (A). [Hint: Because if third zero is $\alpha$, sum of the zeroes $= \alpha+0+0=\frac{-b}{a}$]

EXERCISE 2.1

1. If one of the zeroes of the quadratic polynomial $(k-1)x^{2}+kx+1$ is -3, then the value of k is: (A) $\frac{4}{3}$ (B) $\frac{-4}{3}$ (C) $\frac{2}{3}$ (D) $\frac{-2}{3}$
2. A quadratic polynomial, whose zeroes are -3 and 4, is: (A) $x^{2}-x+12$ (B) $x^{2}+x+12$ (C) $\frac{x^{2}}{2}-\frac{x}{2}-6$ (D) $2x^{2}+2x-24$
3. If the zeroes of the quadratic polynomial $x^2 + (a + 1)x + b$ are 2 and -3, then: (A) $a=-7, b=-1$ (B) $a=5, b=-1$ (C) $a=2, b=-6$ (D) $a=0, b=-6$
4. The number of polynomials having zeroes as -2 and 5 is: (A) 1 (B) 2 (C) 3 (D) more than 3
5. Given that one of the zeroes of the cubic polynomial $ax^{3}+bx^{2}+cx+d$ is zero, the product of the other two zeroes is: (A) $-\frac{c}{a}$ (B) $\frac{c}{a}$ (C) 0 (D) $-\frac{b}{a}$
6. If one of the zeroes of the cubic polynomial $x^{3}+ax^{2}+bx+c$ is -1, then the product of the other two zeroes is: (A) $b-a+1$ (B) $b-a-1$ (C) $a-b+1$ (D) $a-b-1$
7. The zeroes of the quadratic polynomial $x^{2}+99x+127$ are: (A) both positive (B) both negative (C) one positive and one negative (D) both equal
8. The zeroes of the quadratic polynomial $x^{2}+kx+k, k\ne0$: (A) cannot both be positive (B) cannot both be negative (C) are always unequal (D) are always equal
9. If the zeroes of the quadratic polynomial $ax^{2}+bx+c, c\ne0$ are equal, then: (A) c and a have opposite signs (B) c and b have opposite signs (C) c and a have the same sign (D) c and b have the same sign
10. If one of the zeroes of a quadratic polynomial of the form $x^{2}+ax+b$ is the negative of the other, then it: (A) has no linear term and the constant term is negative. (B) has no linear term and the constant term is positive. (C) can have a linear term but the constant term is negative. (D) can have a linear term but the constant term is positive.
11. Which of the following is not the graph of a quadratic polynomial?

(C) Short Answer Questions with Reasoning

Sample Question 1: Can $x-1$ be the remainder on division of a polynomial p(x) by $2x+3$? Justify your answer.

Solution: No, since degree $(x-1)=1=$ degree $(2x+3)$

Sample Question 2: Is the following statement True or False? Justify your answer. If the zeroes of a quadratic polynomial $ax^{2}+bx+c$ are both negative, then a, b and c all have the same sign.

Solution: True, because sum of the zeroes $= -\frac{b}{a} < 0$, so that $\frac{b}{a}> 0$. Also the product of the zeroes $= \frac{c}{a} > 0$.

EXERCISE 2.2

1. Answer the following and justify:

1. Can $x^2-1$ be the quotient on division of $x^{6}+2x^{3}+x-1$ by a polynomial in x of degree 5?
2. What will the quotient and remainder be on division of $ax^{2}+bx+c$ by $px^{3}+qx^{2}+rx+s, p\ne0$?
3. If on division of a polynomial $p(x)$ by a polynomial $g(x)$, the quotient is zero, what is the relation between the degrees of $p(x)$ and $g(x)$?
4. If on division of a non-zero polynomial $p(x)$ by a polynomial $g(x)$, the remainder is zero, what is the relation between the degrees of $p(x)$ and $g(x)$?
5. Can the quadratic polynomial $x^{2}+kx+k$ have equal zeroes for some odd integer $k>1$?

2. Are the following statements 'True' or 'False'? Justify your answers.

1. If the zeroes of a quadratic polynomial $ax^{2}+bx+c$ are both positive, then a, b and c all have the same sign.
2. If the graph of a polynomial intersects the x-axis at only one point, it cannot be a quadratic polynomial.
3. If the graph of a polynomial intersects the x-axis at exactly two points, it need not be a quadratic polynomial.
4. If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
5. If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign.
6. If all three zeroes of a cubic polynomial $x^{3}+ax^{2}-bx+c$ are positive, then at least one of a, b and c is non-negative.
7. The only value of k for which the quadratic polynomial $kx^{2}+x+k$ has equal zeros is $\frac{1}{2}$.

(D) Short Answer Questions

Sample Question 1: Find the zeroes of the polynomial $x^{2}+\frac{1}{6}x-2$, and verify the relation between the coefficients and the zeroes of the polynomial.

Solution: $x^{2}+\frac{1}{6}x-2=\frac{1}{6}(6x^{2}+x-12)=\frac{1}{6}[6x^{2}+9x-8x-12]$

$=\frac{1}{6}[3x(2x+3)-4(2x+3)]=\frac{1}{6}(3x-4)(2x+3)$

Hence, $\frac{4}{3}$ and $-\frac{3}{2}$ are the zeroes of the given polynomial.

The sum of zeroes $=\frac{4}{3}+-\frac{3}{2}=\frac{-1}{6}=-\frac{\text{Coefficient of } x}{\text{Coefficient of } x^{2}}$

and the product of zeroes $=\frac{4}{3}\times\frac{-3}{2}=-2=\frac{\text{Constant term}}{\text{Coefficient of } x^{2}}$

EXERCISE 2.3

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:

1. $4x^{2}-3x-1$
2. $3x^{2}+4x-4$
3. $5t^{2}+12t+7$
4. $t^{3}-2t^{2}-15t$
5. $2x^{2}+\frac{7}{2}x+\frac{3}{4}$
6. $4x^{2}+5\sqrt{2}x-3$
7. $2s^{2}-(1+2\sqrt{2})s+\sqrt{2}$
8. $v^{2}+4\sqrt{3}v-15$
9. $y^{2}+\frac{3}{2}\sqrt{5}y-5$
10. $7y^{2}-\frac{11}{3}y-\frac{2}{3}$

(E) Long Answer Questions

Sample Question 1: Find a quadratic polynomial, the sum and product of zeroes are $\sqrt{2}$ and $-\frac{3}{2}$, respectively. Also find its zeroes.

Solution: A quadratic polynomial, the sum and product of whose zeroes are $\sqrt{2}$ and $-\frac{3}{2}$ is $x^{2}-\sqrt{2}x-\frac{3}{2}$

$x^{2}-\sqrt{2}x-\frac{3}{2}=\frac{1}{2}[2x^{2}-2\sqrt{2}x-3]$

$=\frac{1}{2}[2x^{2}+\sqrt{2}x-3\sqrt{2}x-3] = \frac{1}{2}[\sqrt{2}x(\sqrt{2}x+1)-3(\sqrt{2}x+1)]$

$=\frac{1}{2}[\sqrt{2}x+1][\sqrt{2}x-3]$

Hence, the zeroes are $-\frac{1}{\sqrt{2}}$ and $\frac{3}{\sqrt{2}}$

Sample Question 2: 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 $x^{3}+2x^{2}+kx-18$.

Solution: Let $p(x)=x^{3}+2x^{2}+kx+3$

Then, $p(3)=3^{3}+2\times3^{2}+3k+3=21$

i.e., $3k=-27$ which implies $k=-9$

Hence, the given polynomial will become $x^{3}+2x^{2}-9x+3$.

Performing long division: $x^{3}+2x^{2}-9x+3 = (x^{2}+5x+6)(x-3)+21$

Now, $x^{3}+2x^{2}-9x-18=(x-3)(x^{2}+5x+6) = (x-3)(x+2)(x+3)$

So, the zeroes of $x^{3}+2x^{2}+kx-18$ are 3, -2, -3.

EXEMPLAR PROBLEMS

1. 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) $\frac{-8}{3}, \frac{4}{3}$
• (ii) $\frac{21}{8}, \frac{5}{16}$
• (iii) $-2\sqrt{3}, -9$
• (iv) $-\frac{3}{2\sqrt{5}}, -\frac{1}{2}$

2. Given that the zeroes of the cubic polynomial $x^{3}-6x^{2}+3x+10$ are of the form $a, a+b, a+2b$ for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.

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

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

5. Given that $x-\sqrt{5}$ is a factor of the cubic polynomial $x^{3}-3\sqrt{5}x^{2}+13x-3\sqrt{5}$, find all the zeroes of the polynomial.

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