Polynomials: Class 10 Important Questions with Answers

HOTS (higher-order) practice

H1. If one zero of $x^2-6x+k$ is twice the other, find $k$.

H2. If $\alpha ,\beta$ are zeroes of $x^2-5x+6$, find a polynomial whose zeroes are $2\alpha$ and $2\beta$.

H3. Find the zeroes of $x^2-(\sqrt{3}+1)x+\sqrt{3}$.

H4. For what value of $k$ is $-4$ a zero of $p(x)=x^2-x-(2k+2)$? Find the other zero.

H5. The graph of $y=a{x}^2+bx+c$ opens downward and touches the x-axis at one point. What can you say about $a$ and $b^2-4ac$?

Case-based practice

20 MCQs (1 mark each)

1. The degree of $5x^3-2x^2+7$ is
   (a) 5 (b) 3 (c) 2 (d) 7
2. Which is a polynomial?
   (a) $x+\frac{1}{x}$ (b) $\sqrt{x}+2$ (c) $x^2-\sqrt{3}x$ (d) $x^{-2}+1$
3. The zero of $3x-6$ is
   (a) 3 (b) −2 (c) 2 (d) 6
4. Sum of zeroes of $2x^2-8x+5$ is
   (a) −4 (b) 4 (c) $\frac{5}{2}$ (d) 8
5. Product of zeroes of $3x^2+5x-2$ is
   (a) $-\frac{2}{3}$ (b) $\frac{2}{3}$ (c) $-\frac{5}{3}$ (d) −2
6. A quadratic with zeroes 2 and −3 is
   (a) $x^2-x-6$ (b) $x^2+x-6$ (c) $x^2+x+6$ (d) $x^2-5x+6$
7. If the graph of $p(x)$ cuts the x-axis at 3 points, the number of zeroes is
   (a) 1 (b) 2 (c) 3 (d) 0
8. If one zero of $x^2+kx+6$ is 2, then $k$ is
   (a) 5 (b) −5 (c) 3 (d) −3
9. If $\alpha ,\beta$ are zeroes of $x^2-3x+2$, then ${\alpha }^2+{\beta }^2$ is
   (a) 5 (b) 9 (c) 13 (d) 1
10. If the zeroes of $a{x}^2+bx+c$ are reciprocals, then
    (a) $a=b$ (b) $b=c$ (c) $a=c$ (d) $a=-c$
11. The zeroes of $x^2-9$ are
    (a) 9, −9 (b) 3, 3 (c) 3, −3 (d) 0, 9
12. If the sum of zeroes of $k{x}^2+2x+3k$ equals their product, $k$ is
    (a) $\frac{2}{3}$ (b) $-\frac{2}{3}$ (c) $\frac{3}{2}$ (d) $-\frac{3}{2}$
13. A polynomial of degree 3 has at most
    (a) 1 zero (b) 2 zeroes (c) 3 zeroes (d) 4 zeroes
14. If $\alpha ,\beta$ are zeroes of $x^2+5x+6$, then $\frac{1}{\alpha }+\frac{1}{\beta }$ is
    (a) $\frac{5}{6}$ (b) $-\frac{5}{6}$ (c) $\frac{6}{5}$ (d) −5
15. The graph of $y=-x^2+4$ opens
    (a) upward (b) downward (c) left (d) right
16. If both zeroes of $x^2+px+q$ are 0, then
    (a) p = 0 only (b) q = 0 only (c) p = q = 0 (d) p = 1
17. The value of $p(x)=2x^2-3x+1$ at $x=-1$ is
    (a) 0 (b) 6 (c) 2 (d) −4
18. For $x^3-6x^2+11x-6$, the sum of zeroes is
    (a) −6 (b) 6 (c) 11 (d) −11
19. If one zero of $x^2-4x+k$ is 3 times the other, $k$ is
    (a) 1 (b) 3 (c) 4 (d) 12
20. $x^2+2x+5$ has
    (a) 2 real zeroes (b) 1 real zero (c) no real zero (d) 3 zeroes

Answer key

1 (b) · 2 (c) · 3 (c) · 4 (b) · 5 (a) · 6 (b) · 7 (c) · 8 (b) · 9 (a) · 10 (c) · 11 (c) · 12 (b) · 13 (c) · 14 (b) · 15 (b) · 16 (c) · 17 (b) · 18 (b) · 19 (b) · 20 (c)

Hints: Q8 $4+2k+6=0$. Q9 $9-4=5$. Q12 $-\frac{2}{k}=3\Rightarrow k=-\frac{2}{3}$. Q14 $\frac{-5}{6}$. Q17 $2+3+1=6$. Q19 zeroes 1, 3 ⇒ k = 3. Q20 $4-20<0$.

Assertion–Reason (10)

(a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not the correct explanation of A. (c) A is true but R is false. (d) A is false but R is true.

1. A: $x^2-4$ has zeroes 2 and −2. R: The sum of the zeroes of $a{x}^2+bx+c$ is $-\frac{b}{a}$.
2. A: $x^2+1$ has no real zero. R: $x^2+1>0$ for every real $x$.
3. A: If 1 and −2 are zeroes, the polynomial is $x^2+x-2$. R: $p(x)=x^2-(\alpha +\beta )x+\alpha \beta$.
4. A: A cubic polynomial can have 4 zeroes. R: A polynomial of degree $n$ has at most $n$ zeroes.
5. A: The zeroes of $4u^2+8u$ are 0 and −2. R: If the constant term is 0, one zero is 0.
6. A: $\sqrt{x}+3$ is a polynomial. R: The powers of the variable in a polynomial are whole numbers.
7. A: The graph of $y=x^2-6x+9$ touches the x-axis at one point. R: $x^2-6x+9=(x-3{)}^2$ has two equal zeroes.
8. A: For $2x^2-5x+3$, the product of zeroes is $\frac{3}{2}$. R: Product of zeroes $=\frac{c}{a}$.
9. A: If the zeroes of $x^2+bx+7$ are reciprocals, then $7=1$. R: Reciprocal zeroes need $a=c$.
10. A: Every linear polynomial has exactly one zero. R: The graph of a linear polynomial is a straight line not parallel to the x-axis.

Key: 1 (b) · 2 (a) · 3 (a) · 4 (d) · 5 (a) · 6 (d) · 7 (a) · 8 (a) · 9 (d) · 10 (a)

Q9: reciprocal zeroes would need $a=c$, i.e. $1=7$, which is impossible; so A (as a conclusion) is false while R is true ⇒ (d).

Very short answer (1 mark each)

1. Write a quadratic polynomial with sum of zeroes 3 and product 2. Ans. $x^2-3x+2$
2. If $p(x)=x^2-4x+3$, find $p(1)$. Ans. 0 (so 1 is a zero)
3. How many zeroes can a polynomial of degree 4 have at most? Ans. 4
4. Find the zeroes of $x^2+7x$. Ans. 0, −7
5. If $\alpha ,\beta$ are zeroes of $x^2-7x+10$, find $\alpha \beta -(\alpha +\beta )$. Ans. $10-7=3$
6. Write the polynomial whose zeroes are $\sqrt{3}$ and $-\sqrt{3}$. Ans. $x^2-3$
7. If the graph of $p(x)$ is a ∩ meeting the x-axis at two points, what is the sign of $a$? Ans. negative
8. If 2 is a zero of $x^2-3x+k$, find $k$. Ans. $4-6+k=0\Rightarrow k=2$

Short answer practice (2 marks each)

1. Find a quadratic polynomial whose zeroes are $3+\sqrt{2}$ and $3-\sqrt{2}$.

2. If $\alpha ,\beta$ are zeroes of $x^2-8x+k$ and ${\alpha }^2+{\beta }^2=40$, find $k$.

3. If the sum of the zeroes of $(a+1)x^2+(2a+3)x+(3a+4)$ is −1, find $a$.

Short answer practice (3 marks each)

4. Find the zeroes of $6x^2-7x-3$ and verify the relationship with the coefficients.

5. If $\alpha ,\beta$ are zeroes of $2x^2-5x+7$, find a polynomial whose zeroes are $2\alpha +3\beta$ and $3\alpha +2\beta$.

6. If one zero of $3x^2-8x+2k+1$ is seven times the other, find the zeroes and $k$.

7. Verify that $3,-1,-\frac{1}{3}$ are the zeroes of $3x^3-5x^2-11x-3$ and verify the relationships.

Long answer (5 marks)

8. $\alpha ,\beta$ are zeroes of $x^2-2x-8$. (i) Find $\alpha$ and $\beta$ by factorising. (ii) Without using their values, find ${\alpha }^2+{\beta }^2$ and ${\alpha }^3+{\beta }^3$. (iii) Form the quadratic whose zeroes are ${\alpha }^2$ and ${\beta }^2$.