Polynomials · Class 10 Mathematics: 170 MCQs with answers

Q1. Which of these is a polynomial?

Q2. The degree of $5x^3 - 4x^2 + x - 2$ is

Q3. A polynomial of degree 2 is called

Q4. The general form of a linear polynomial is

Q5. For $p(x) = x^2 - 3x - 4$, the value of $p(0)$ is

Q6. For $p(x) = 2x + 7$, $p(3)$ equals

Q7. A real number k is a zero of p(x) when

Q8. The zero of the polynomial $3x - 12$ is

Q9. The zero of $ax + b$ ($a \neq 0$) is

Q10. How many zeroes does a linear polynomial have?

Q11. The degree of the polynomial $3 - x^2 + x^3$ is

Q12. Which of these is a cubic polynomial?

Q13. Assertion (A): $x^2 + \frac{1}{x}$ is not a polynomial. Reason (R): In a polynomial every power of the variable must be a whole number.

Q14. The coefficient of $x^2$ in $5 - 3x + 7x^2 - x^3$ is

Q15. The constant term of $p(x) = 4x^3 - x + 6$ is

Q16. State which of these are polynomials in x, giving a reason: (i) $4x^2 - 3x + 7$ (ii) $\frac{2}{x} + 5$ (iii) $\sqrt{3}x^2 - 2x$ (iv) $x^{3/2} + 1$.

Q17. For $p(x) = x^2 - 3x - 4$, find p(2), p(0) and p(−1), and say which of these numbers is a zero.

Q18. Find the zero of each linear polynomial: (i) $2x + 3$ (ii) $5x - 20$ (iii) $-3x + 9$.

Q19. Verify whether 2 and −1 are zeroes of $p(x) = x^2 - x - 2$.

Q20. The zeroes of a polynomial are the x-coordinates of the points where its graph meets

Q21. The graph of a linear polynomial is

Q22. The graph of $y = ax^2 + bx + c$ with $a > 0$

Q23. A graph meets the x-axis at exactly 2 points. The number of zeroes is

Q24. The graph of a quadratic polynomial touches the x-axis at exactly one point. The polynomial has

Q25. A parabola lies entirely above the x-axis. The number of real zeroes is

Q26. The graph of $y = 2x + 3$ cuts the x-axis at

Q27. The point where a graph meets the y-axis gives

Q28. How many times does the graph of a polynomial of degree 4 meet the x-axis, at most?

Q29. Assertion (A): The graph of a linear polynomial cuts the x-axis at exactly one point. Reason (R): A linear polynomial has exactly one zero.

Q30. Which curve can meet the x-axis three times?

Q31. Where does $y = x^2$ meet the x-axis?

Q32. What is the geometrical meaning of a zero of a polynomial? Illustrate with $p(x) = 2x + 3$.

Q33. Describe the three ways in which the graph of a quadratic polynomial can meet the x-axis, and give the number of zeroes in each case.

Q34. For the zeroes α and β of $ax^2 + bx + c$, the value of $\alpha + \beta$ equals

Q35. For the same polynomial, $\alpha\beta$ equals

Q36. The sum of the zeroes of $x^2 - 7x + 12$ is

Q37. The product of the zeroes of $x^2 - 7x + 12$ is

Q38. The sum of the zeroes of $2x^2 - 8x + 6$ is

Q39. A quadratic polynomial whose zeroes have sum 5 and product 6 is

Q40. A quadratic polynomial whose zeroes have sum −3 and product 2 is

Q41. The zeroes of $x^2 + 7x + 10$ are

Q42. In splitting the middle term of $ax^2 + bx + c$, the two numbers chosen must have product

Q43. The product of the zeroes of $4x^2 + 8x$ is

Q44. Assertion (A): For $x^2 - 3$, the sum of the zeroes is 0. Reason (R): The sum of the zeroes equals $-\frac{b}{a}$, and here $b = 0$.

Q45. The zeroes of $x^2 - 15$ are

Q46. Find the zeroes of $x^2 - 2x - 8$ and verify the relationship between the zeroes and the coefficients.

Q47. Find a quadratic polynomial whose zeroes have sum $\frac{1}{4}$ and product −1.

Q48. For $p(x) = x^2 - 3x - 4$, which number is a zero?

Q49. If $p(x) = 5x^2 - 4x + 2$, then $p(1)$ is

Q50. If $p(y) = y^3 - 3y + 5$, then $p(-2)$ is

Q51. Which of these has 2 as a zero?

Q52. If $(k - 2)x^2 + 3x + 5$ is a linear polynomial, then k is

Q53. If 4 is a zero of $p(x) = x^2 - 6x + k$, then k is

Q54. If $-2$ is a zero of $p(x) = 2x^2 + 3x + m$, then m is

Q55. The number of zeroes a cubic polynomial can have at most is

Q56. Which quadratic polynomial has NO real zero?

Q57. The zero of $p(x) = 7x$ is

Q58. If $p(x) = x^2 - 1$, then $p(1) + p(-1)$ equals

Q59. A polynomial whose value is 0 for x = 5 must have

Q60. Assertion (A): $x^2 + 1$ has no real zero. Reason (R): $x^2 + 1$ is a quadratic polynomial.

Q61. Assertion (A): Every quadratic polynomial has exactly two zeroes. Reason (R): A polynomial of degree n has at most n zeroes.

Q62. Which is NOT a polynomial in x?

Q63. The value of $p(x) = x^3 - 2x^2 + x$ at $x = 2$ is

Q64. Which polynomial has 0 as one of its zeroes?

Q65. A polynomial p(x) has $p(3) = 0$. Which statement must be true?

Q66. If $p(x) = x^2 - 9$, then the zeroes of p(x) are

Q67. If 3 is a zero of $p(x) = x^2 - 4x + k$, find k and then the other zero.

Q68. If $p(x) = (k - 3)x^2 + 5x - 2$ is a linear polynomial, find k and the zero of the resulting polynomial.

Q69. Give one example each of a quadratic polynomial with (i) two zeroes (ii) exactly one zero (iii) no real zero, and state the zeroes where they exist.

Q70. If $p(x) = 2x^3 - 3x^2 + 4x - 5$, find $p(1)$ and $p(-1)$.

Q71. (a) Define the degree of a polynomial and name the polynomials of degree 1, 2 and 3. (b) For $p(x) = x^3 - 2x^2 - x + 2$, find p(1), p(2) and p(−1). (c) Write down the zeroes of p(x) from your answers in (b), and say how you know there are no others.

Q72. A shopkeeper's daily profit, in rupees, on selling x kites is $P(x) = x^2 - 12x + 20$. (a) Find the profit when 5 kites and when 10 kites are sold. (b) For which numbers of kites is the profit zero? (c) What does a negative value of P(x) mean, and give one value of x where it happens?

Q73. A graph cuts the x-axis at 1 point and has no turning point. The polynomial is most likely

Q74. A parabola opens downwards and cuts the x-axis at two points. Which is true?

Q75. Which of these points lies on the curve $y = x^2 - 3x - 4$?

Q76. For $y = x^2 - 3x - 4$, the value of y at $x = 1$ is

Q77. A graph meets the x-axis at 3 points and the y-axis at 1 point. How many zeroes does the polynomial have?

Q78. Which polynomial's graph TOUCHES the x-axis without crossing it?

Q79. A cubic polynomial's graph meets the x-axis at exactly 1 point. Its number of zeroes is

Q80. If the graph of a quadratic polynomial does not meet the x-axis at all, then

Q81. The zeroes of the polynomial whose graph cuts the x-axis at $(-3, 0)$ and $(5, 0)$ are

Q82. Which of these could NOT be the number of zeroes of a quadratic polynomial?

Q83. Assertion (A): A graph that meets the x-axis at four points belongs to a polynomial of degree at least 4. Reason (R): A polynomial of degree $n$ has at least $n$ zeroes.

Q84. Assertion (A): The point $(0, 5)$ on the graph of $y = p(x)$ means 5 is a zero of p(x). Reason (R): The zeroes of p(x) are the x-coordinates of the points where the graph meets the x-axis.

Q85. A parabola passes through $(2, 0)$ and $(6, 0)$ and opens upwards. Between $x = 2$ and $x = 6$ the curve is

Q86. How many zeroes does the polynomial have if its graph is a straight line lying ALONG the x-axis?

Q87. For which polynomial does the graph meet the x-axis at exactly one point and lie above it everywhere else?

Q88. Which of these graphs must cross the x-axis at least once?

Q89. At which points does $y = -x^2 + 9$ cross the x-axis?

Q90. A graph passes through $(0, 0)$. Which statement is certain?

Q91. The number of turning points of a parabola is

Q92. A graph of $y = p(x)$ meets the x-axis at 3 points and has 2 turning points. How many zeroes has p(x), and what can you say about its degree?

Q93. For $p(x) = x^2 - 3x - 4$, complete the values at $x = -2, -1, 0, 1, 2, 3, 4$ and state the zeroes from the table.

Q94. The graph of a quadratic polynomial passes through $(0, 6)$ and lies completely above the x-axis. How many zeroes does it have, and what is the constant term?

Q95. Explain why the graph of a quadratic polynomial can never meet the x-axis at three points.

Q96. State whether each graph could belong to a quadratic polynomial: (i) a curve cutting the x-axis at 2 points (ii) a straight line (iii) a curve meeting the x-axis at 3 points.

Q97. (a) State the geometrical meaning of the zeroes of a polynomial. (b) Draw a rough sketch, described in words, of a quadratic polynomial with (i) two distinct zeroes (ii) two equal zeroes (iii) no real zero. (c) In each case state how many times the curve meets the x-axis.

Q98. A stone is thrown upwards from a cliff. Its height above the sea, in metres, after t seconds is $h(t) = -5t^2 + 10t + 40$, and the graph of h against t is a parabola. (a) Which way does the parabola open, and why? (b) What is the height at $t = 0$, and what does the point $(0, 40)$ mean? (c) Show that $t = 4$ is a zero of h(t) and say what it means. Why is $t = -2$, the other zero, of no use here?

Q99. The zeroes of $6x^2 - 7x - 3$ are

Q100. If the sum of the zeroes of $kx^2 - 3x + 5$ is 1, then k is

Q101. If the product of the zeroes of $2x^2 + 5x + k$ is 3, then k is

Q102. Let α and β be the zeroes of $x^2 - 5x + 6$. Then $\alpha^2 + \beta^2$ is

Q103. Given the zeroes α and β of $x^2 - 6x + 8$, the value of $\frac{1}{\alpha} + \frac{1}{\beta}$ is

Q104. A quadratic polynomial whose zeroes are 3 and −5 is

Q105. Which quadratic polynomial has zeroes whose sum is 0 and product is −9?

Q106. The zeroes of $x^2 - 2x - 8$ are

Q107. For $4x^2 - 4x + 1$, the zeroes are

Q108. If one zero of $x^2 - 4x + k$ is 3, the other zero is

Q109. If the zeroes of a quadratic are equal, then

Q110. How many quadratic polynomials have zeroes 2 and 3?

Q111. Assertion (A): $2x^2 - 10x + 12$ and $x^2 - 5x + 6$ have different zeroes. Reason (R): Multiplying a polynomial by a non-zero constant does not change its zeroes.

Q112. Assertion (A): The sum of the zeroes of $3x^2 - 6x + 2$ is 6. Reason (R): The sum of the zeroes of $ax^2 + bx + c$ is $-\frac{b}{a}$.

Q113. Two numbers have sum 27 and product 182. The numbers are

Q114. For $3x^2 - x - 4$, the sum and product of the zeroes are

Q115. If the zeroes of a quadratic are 1 and 1, the polynomial could be

Q116. For $x^2 + 4x + 4$, the sum and product of the zeroes are

Q117. If α and β are the zeroes of $x^2 - 2x - 15$, then $\alpha\beta + \alpha + \beta$ equals

Q118. Find the zeroes of $6x^2 - 3 - 7x$ and verify both relations.

Q119. Find a quadratic polynomial whose zeroes are $\sqrt{2}$ and $-\sqrt{2}$.

Q120. If α and β are the zeroes of $x^2 - 5x + 4$, find $\alpha^2 + \beta^2$ without finding the zeroes.

Q121. If α and β are the zeroes of $2x^2 - 7x + 3$, find $\frac{1}{\alpha} + \frac{1}{\beta}$.

Q122. One zero of $x^2 - 9x + k$ is 4. Find the other zero and k.

Q123. Two numbers have a sum of 17 and a product of 60. Form a quadratic polynomial with these numbers as zeroes and find them.

Q124. (a) State the relationship between the zeroes and the coefficients of $ax^2 + bx + c$. (b) Find the zeroes of $4x^2 + 8x$ and verify both relations. (c) Write a quadratic polynomial whose zeroes are the squares of the zeroes found in (b).

Q125. (a) Find a quadratic polynomial whose zeroes have sum −3 and product 2, and find its zeroes. (b) Show that $2x^2 + 6x + 4$ has the same zeroes, and explain why. (c) How many quadratic polynomials have these zeroes? Justify your answer.

Q126. If 2 and −3 are both zeroes of $p(x) = x^2 + ax + b$, then a and b are

Q127. For which value of k does $p(x) = kx^2 - 3x + k$ have 1 as a zero?

Q128. If $p(x) = ax + b$ has zero 4 and $p(0) = -8$, then a is

Q129. A polynomial p(x) satisfies $p(2) = 0$ and $p(-5) = 0$. Its lowest possible degree is

Q130. If the value of $x^2 - kx + 6$ at $x = 2$ is 0, then the other zero is

Q131. $p(x) = x^2 + 2x - 3$. How many of the numbers −3, −1, 1 and 3 are zeroes of p(x)?

Q132. If $p(x) = x^{2} - x$, which statement is true?

Q133. Assertion (A): If $p(x) = (x - 1)(x - 2)(x - 3)$, then p(x) has three zeroes. Reason (R): A cubic polynomial has at most three zeroes.

Q134. The degree of the product of a quadratic and a cubic polynomial is

Q135. If $p(x) = 2x^3 - 3x^2 + 4x - 5$, then $p(1) - p(0)$ equals

Q136. The polynomial $p(x) = ax + b$ has $p(2) = 7$ and $p(-1) = -2$. Find a and b, and hence the zero of p(x).

Q137. Show that $x = -\frac{b}{a}$ is the only zero of the linear polynomial $ax + b$, $a \neq 0$.

Q138. (a) If 2 is a zero of $p(x) = x^2 + kx - 6$, find k. (b) Find the other zero. (c) Write a quadratic polynomial whose zeroes are the same two numbers but which is not equal to p(x).

Q139. (a) Explain, with an example each, why $\frac{1}{x} + 2$ and $\sqrt{x} - 1$ are not polynomials while $\sqrt{5}x^2 - x$ is. (b) Find the degree of $(x^2 + 1)(x^3 - x)$ without multiplying it out, and give a reason. (c) Find p(0) and p(2) for $p(x) = (x^2 + 1)(x - 2)$, and name one zero of p(x).

Q140. The graph of a polynomial meets the x-axis at 3 points and the curve rises at both ends. The least possible degree is

Q141. A quadratic polynomial has exactly one zero. Which statement must be true?

Q142. $y = p(x)$ lies entirely below the x-axis. Which is true about p(x)?

Q143. A parabola cuts the x-axis at $(-2, 0)$ and $(3, 0)$ and passes through $(0, -6)$. A polynomial that fits is

Q144. Which pair of graphs has the SAME number of zeroes?

Q145. Assertion (A): A cubic polynomial always has three zeroes. Reason (R): The graph of a cubic polynomial always meets the x-axis.

Q146. A graph touches the x-axis at $x = 1$ and cuts it at $x = -2$. The least possible degree is

Q147. The graph of $y = p(x)$ is a parabola with its lowest point at $(2, 0)$. Then p(x) has

Q148. For a quadratic whose graph opens downwards and touches the x-axis, the value of p(x) is

Q149. A curve meets the x-axis at 5 points. The polynomial's degree is at least

Q150. Assertion (A): The graph of $y = x^2 + 1$ has no point on the x-axis. Reason (R): $x^2 + 1 \ge 1$ for every real number x.

Q151. The lowest point of a parabola is $(3, 0)$. How many zeroes does the polynomial have, and what are they?

Q152. Sketch in words the shape of $y = x^3 - 4x$ and give its zeroes.

Q153. Two parabolas have the same shape, but one meets the x-axis twice and the other not at all. Give an example of each and explain the difference.

Q154. For each description, state the number of zeroes and give the least possible degree, with a reason: (a) a straight line crossing the x-axis once (b) a curve touching the x-axis at one point and lying above it elsewhere (c) a curve cutting the x-axis at −1, 1 and 4 (d) a curve lying entirely below the x-axis (e) a curve touching the axis at 2 and cutting it at −3.

Q155. A parabola cuts the x-axis at $(-2, 0)$ and $(3, 0)$ and passes through $(0, -6)$. (a) Write down the zeroes of the polynomial. (b) Find a quadratic polynomial with these zeroes whose graph passes through $(0, -6)$. (c) Does the parabola open upwards or downwards? Give a reason.

Q156. Suppose α and β are the zeroes of $x^2 - 7x + 10$. Then $(\alpha - \beta)^2$ is

Q157. For the zeroes α and β of $x^2 + 5x + 6$, a quadratic whose zeroes are $2\alpha$ and $2\beta$ is

Q158. If one zero of $2x^2 - 8x + k$ is three times the other, then k is

Q159. If the zeroes of $x^2 + px + q$ are equal in magnitude but opposite in sign, then

Q160. If α and β are the zeroes of $x^2 - 4x + 3$, then $\alpha^3 + \beta^3$ is

Q161. A quadratic polynomial whose zeroes are the reciprocals of the zeroes of $2x^2 - 5x + 3$ is

Q162. If the sum and product of the zeroes of a quadratic are both 6, the polynomial is

Q163. Assertion (A): A quadratic polynomial with zeroes $\sqrt2$ and $-\sqrt2$ is $x^2 - 2$. Reason (R): The sum of the zeroes is 0 and their product is 2.

Q164. The zeroes of $x^2 - 6x + 8$ are α and β with $\alpha > \beta$, then $\alpha - \beta$ is

Q165. For which value of m are the zeroes of $x^2 - mx + 9$ equal?

Q166. A quadratic polynomial with zeroes $\frac{1}{2}$ and $-\frac{3}{2}$ is

Q167. If the sum of the zeroes of $kx^2 + 2x + 3k$ is equal to their product, find k.

Q168. Prove that the sum of the zeroes of $ax^2 + bx + c$ is $-\frac{b}{a}$.

Q169. (a) Find the zeroes of $3x^2 - x - 4$ and verify the relations between the zeroes and the coefficients. (b) If α and β are those zeroes, find $\alpha^2 + \beta^2$. (c) Write a quadratic polynomial whose zeroes are $\alpha + 1$ and $\beta + 1$.

Q170. A rectangular kitchen garden has a perimeter of 34 m and an area of 60 m². (a) Taking the sides as α and β, write down their sum and product. (b) Form a quadratic polynomial with α and β as zeroes, and find the sides. (c) The gardener now wants a perimeter of 34 m and an area of 80 m². Show that this is not possible.