Polynomials · Class 10 Mathematics: 62 flashcards
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1. What is a polynomial?
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An expression in which the variable appears only with whole-number powers.
2. Is $\frac{1}{x+1}$ a polynomial?
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No — it is $(x+1)^{-1}$, and −1 is not a whole-number power.
3. Is $3\sqrt{x} + 2$ a polynomial?
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No — $\sqrt{x} = x^{1/2}$.
4. What is the degree of a polynomial?
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The highest power of the variable.
5. Degree of $7u^6 - \frac{3}{2}u^4 + 4u^2 + u - 8$?
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6
6. General form of a quadratic polynomial?
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$ax^2 + bx + c$, with a, b, c real and $a \neq 0$.
7. General form of a cubic polynomial?
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$ax^3 + bx^2 + cx + d$, $a \neq 0$.
8. What does p(k) mean?
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The value of the polynomial when x is replaced by k.
9. For $p(x) = x^2 - 3x - 4$, find p(2).
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$4 - 6 - 4 = -6$
10. For $p(x) = x^2 - 3x - 4$, find p(−1).
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$1 + 3 - 4 = 0$, so −1 is a zero.
11. When is k a zero of p(x)?
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When $p(k) = 0$.
12. Zeroes of $x^2 - 3x - 4$?
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−1 and 4
13. Zero of the linear polynomial $ax + b$?
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$-\frac{b}{a}$
14. Zero of $2x + 3$?
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$-\frac{3}{2}$
15. Zero of $5x - 20$?
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4
16. How many zeroes can a polynomial of degree n have?
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At most n.
17. Give a quadratic polynomial with no real zero.
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$x^2 + 1$ — its value is at least 1 for every real x.
18. Difference between a zero and a root?
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None in value: zeroes belong to a polynomial, roots to an equation.
19. Does a non-zero constant polynomial have a zero?
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No — 5 is never 0.
20. If $(k-1)x^2 + 3x + 2$ is linear, what is k?
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k = 1, so that the $x^2$ term disappears.
21. Is $\sqrt{2}x^2 - 3x + \frac{1}{7}$ a polynomial?
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Yes — the coefficients may be surds or fractions; only the powers must be whole numbers.
22. What does a zero of p(x) look like on the graph of y = p(x)?
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The x-coordinate of a point where the graph meets the x-axis.
23. How many times does the graph of a linear polynomial cross the x-axis?
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Exactly once, at $\left(-\frac{b}{a}, 0\right)$.
24. What is the graph of a quadratic polynomial called?
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A parabola.
25. When does a parabola open upwards?
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When $a > 0$ in $ax^2 + bx + c$.
26. When does a parabola open downwards?
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When $a < 0$.
27. What does it mean when a parabola CUTS the x-axis at two points?
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The polynomial has two distinct zeroes.
28. What does it mean when a parabola TOUCHES the x-axis?
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The two zeroes are equal — one value, repeated.
29. What does it mean when a parabola misses the x-axis?
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The polynomial has no real zero.
30. How many zeroes can a quadratic have?
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Two, one (repeated) or none.
31. What are the zeroes of $y = x^2 - 3x - 4$ on its graph?
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−1 and 4, where the parabola cuts the x-axis.
32. The graph meets the y-axis at $(0, -4)$. Is −4 a zero?
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No — that is p(0), the constant term.
33. A graph has two turning points and meets the x-axis three times. How many zeroes?
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Three — only the x-axis meetings count.
34. What are the zeroes of $y = x^3 - 4x$?
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−2, 0 and 2.
35. How many zeroes does $y = x^3$ have?
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One, at x = 0, even though its degree is 3.
36. How many zeroes does $y = x^3 - x^2$ have?
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Two: 0 and 1.
37. At most how many points can the graph of a degree-n polynomial meet the x-axis?
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n
38. A graph crosses the x-axis at 3 points. Can the polynomial be quadratic?
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No — a quadratic meets the axis at most twice.
39. Are students asked to plot quadratic graphs in the exam?
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No — NCERT says plotting is not to be done by students or assessed. Reading graphs is.
40. Which point on the graph gives the constant term?
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The y-intercept, $(0, p(0))$.
41. Do $y = x^2 + 2$ and $y = x^2 - 4$ have the same number of zeroes?
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No: none and two. Same shape, different position.
42. If α and β are the zeroes of $ax^2 + bx + c$, what is α + β?
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$-\frac{b}{a}$
43. If α and β are the zeroes of $ax^2 + bx + c$, what is αβ?
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$\frac{c}{a}$
44. Sum of the zeroes of $2x^2 - 8x + 6$?
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$\frac{8}{2} = 4$
45. Product of the zeroes of $2x^2 - 8x + 6$?
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$\frac{6}{2} = 3$
46. Zeroes of $3x^2 + 5x - 2$?
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$\frac13$ and −2, from $(3x - 1)(x + 2)$.
47. How do you build a quadratic with zero-sum S and zero-product P?
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$k[x^2 - Sx + P]$, any $k \neq 0$; usually take k = 1.
48. A quadratic whose zeroes have sum −3 and product 2?
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$x^2 + 3x + 2$
49. Is there only one quadratic with zeroes 2 and 3?
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No — every non-zero multiple of $x^2 - 5x + 6$ has the same zeroes.
50. In splitting the middle term of $ax^2 + bx + c$, the two numbers have product…
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ac, and sum b.
51. Write $\alpha^2 + \beta^2$ using the sum and product.
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$(\alpha + \beta)^2 - 2\alpha\beta$
52. Write $\frac{1}{\alpha} + \frac{1}{\beta}$ using the sum and product.
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$\frac{\alpha + \beta}{\alpha\beta}$
53. Write $(\alpha - \beta)^2$ using the sum and product.
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$(\alpha + \beta)^2 - 4\alpha\beta$
54. For $x^2 - 5x + 6$, find α² + β² without finding the zeroes.
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$25 - 12 = 13$
55. Sum and product of the zeroes of $x^2 - 3$?
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Sum 0, product −3 (zeroes $\pm\sqrt3$).
56. If the sum of the zeroes of $kx^2 - 3x + 5$ is 1, find k.
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$\frac{3}{k} = 1$, so k = 3.
57. Two numbers add to 17 and multiply to 60. Find them.
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5 and 12 — the zeroes of $x^2 - 17x + 60$.
58. Sum of the zeroes of $x^2 + 7x + 10$?
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−7
59. Product of the zeroes of $4x^2 + 8x$?
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0 — the constant term is 0.
60. What must be true for the relations to apply?
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$a \neq 0$ — the polynomial must really be quadratic.
61. Whose formulas are these?
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Vieta's (François Viète, 16th century).
62. If one zero of a quadratic is 0, what does that tell you?
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The constant term c is 0, since the product of zeroes is 0.
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