Mathematics Multiple Choice Questions for Schools on “Geometrical Meaning of Zeros of Polynomial”.
1. The graph of the polynomial 4x2-8x+3 cuts the x-axis at ________ and ________ points.
a) ((frac {3}{4}), 0), ((frac {1}{2}), 0)
b) ((frac {3}{2}), 0), ((frac {1}{2}), 0)
c) ((frac {3}{2}), 0), ((frac {1}{6}), 0)
d) ((frac {7}{2}), 0), ((frac {3}{2}), 0)
Answer: b
Clarification: The graph of the polynomial cuts the x-axis. Only the zeros of the polynomial cut the x-axis.
4x2-8x+3=0
4x2-6x-2x+3=0
2x(2x-3)-1(2x-3)=0
(2x-3)(2x-1)=0
x=(frac {3}{2}, frac {1}{2})
Hence, the graph of the polynomial cuts the x-axis at ((frac {3}{2}), 0) and ((frac {1}{2}), 0).
2. The graph of the polynomial 2x2-8x+5 cuts the y-axis at __________
a) (6, 0)
b) (0, 7)
c) (0, 5)
d) (8, 9)
Answer: c
Clarification: The graph of the polynomial 2x2-8x+5 cuts the y-axis.
Hence, the value of x will be 0.
y(0)=2(0)2-8(0)+5
y=5
The graph cuts the y-axis at (0,5)
3. How many points will the graph of x2+2x+1 will cut the x-axis?
a) 3
b) 1
c) 2
d) 0
Answer: d
Clarification: The graph of x2+2x+1 does not cut the x-axis, because it has imaginary roots.
x2+2x+1=0
x2+x+x+1=0
x(x+1)+(x+1)=0
(x+1)(x+1)=0
x=-1, -1
4. The graph of the quadratic polynomial -x2+x+90 will open upwards.
a) False
b) True
Clarification: The graph of the polynomial will have a downward opening since, a
The graph for the same can be observed here,
5. If the graph of a polynomial cuts the x-axis at 3 points, then the polynomial is ______
a) Linear
b) Quadratic
c) Cubic
d) Biquadratic
Answer: c
Clarification: Since, the graph of the polynomial cuts the x-axis at 3 points, hence, it will be a cubic polynomial. A polynomial is said to be linear, quadratic, cubic or biquadratic according to the degree of the polynomial.
6. What will be the nature of the zeros of a quadratic polynomial if it cuts the x-axis at two different points?
a) Real
b) Distinct
c) Real, Distinct
d) Complex
Answer: c
Clarification: The zeros of the quadratic polynomial cut the x-axis at two different points.
∴ b2 – 4ac ≥ 0
Hence, the nature of the zeros will be real and distinct.
7. The graph of a quadratic polynomial cuts the x-axis at only one point. Hence, the zeros of the quadratic polynomial are equal and real.
a) True
b) False
Answer: a
Clarification: If the graph meets x-axis at one point only, then the quadratic polynomial has coincident zeros. Also, the discriminant of the quadratic polynomial is zero, therefore roots will be real.
8. A real number is called zeros of the polynomial p(x) if _________
a) p(α)=4
b) p(α)=1
c) p(α)≠0
d) p(α)=0
Answer: d
Clarification: A number is called zero of polynomial when it satisfies the equation of the polynomial.
9. If a < 0, then the graph of ax2+bx+c, has a downward opening.
a) True
b) False
Clarification: The leading coefficient of the polynomial is less than zero, hence, it has downward opening. For example, the graph of -x2 is
10. A polynomial is said to be linear, quadratic, cubic or biquadratic according to the degree of the polynomial.
a) False
b) True
Answer: b
Clarification: The degree of the polynomial is the highest of the degree of the polynomial. Hence, a polynomial with highest degree one is linear, two as quadratic and so on.
11. Which of the following is a polynomial?
a) x2+2x+5
b) √x+2x+4
c) x(frac {2}{3})+10x
d) 5x+(frac {5}{x})
Answer: a
Clarification: An expression in the form of (x)=a0+a1x+a2x2+…+anxn, where an≠0, is called a polynomial where a1, a2 … an are real numbers and each power of x is a non-negative integer.
In case of √x+2x+4 , the power of √x is not an integer. Similarly for x(frac {2}{3})+10x, (frac {2}{3}) is a fraction.
Now, 5x+(frac {5}{x}) in this case the power of x is a negative integer. Hence it is not a polynomial.
12. The biquadratic polynomial from the following is ______
a) (x2+3)(x2-3)
b) x2-7
c) x7+x6+x5
d) 5x-3
Answer: a
Clarification: A biquadratic polynomial has highest power 4.
Hence, the polynomial with the highest power as 4 is x4-9 or (x2+3)(x2-3).
13. Which of the following is not a polynomial?
a) x2+5x+10
b) √x+2x+4
c) x10+10x
d) 5x+4
Answer: b
Clarification: An expression in the form of (x)=a0+a1x+a2x2+…+anxn, where an≠0, is called a polynomial where a1, a2 … an are real numbers and each power of x is a non-negative integer.
In case of √x+2x+4, the power of x is not an integer.
Therefore it is not a polynomial.
14. If the zeros of a polynomial are 3 and -5, then they cut the x-axis at ____ and _____ points.
a) (8, 0) and (-4, 0)
b) (3, -3) and (-5, 5)
c) (-3, 0) and (5, 0)
d) (3, 0) and (-5, 0)
Answer: d
Clarification: Since, the zeros of the polynomial are 3 and -5.
Therefore, x = 3 and x = -5 and they cut the x-axis so the y-coordinate will be zero.
Hence, the points it cuts the x-axis will be (3, 0) and (-5, 0).
15. If the graph of the quadratic polynomial is completely above or below the x-axis, then the nature of roots of the polynomial is _____
a) Real and Distinct
b) Distinct
c) Real
d) Complex
Answer: d
Clarification: Since, the graph is completely above or below the x-axis, hence, it has no real roots. If a polynomial has real roots only then it cuts the x-axis. If it lies above or below, the roots are complex in nature.