This set of Mathematics Multiple Choice Questions for Schools focuses on “Geometrical Meaning of Zeros of Polynomial”.

1. The graph of the polynomial 4x^{2}-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)

View Answer

Explanation: The graph of the polynomial cuts the x-axis. Only the zeros of the polynomial cut the x-axis.

4x

^{2}-8x+3=0

4x

^{2}-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 2x^{2}-8x+5 cuts the y-axis at __________

a) (6, 0)

b) (0, 7)

c) (0, 5)

d) (8, 9)

View Answer

Explanation: The graph of the polynomial 2x

^{2}-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 x^{2}+2x+1 will cut the x-axis?

a) 3

b) 1

c) 2

d) 0

View Answer

Explanation: The graph of x

^{2}+2x+1 does not cut the x-axis, because it has imaginary roots.

x

^{2}+2x+1=0

x

^{2}+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 -x^{2}+x+90 will open upwards.

a) False

b) True

View Answer

Explanation: The graph of the polynomial will have a downward opening since, a<0

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

View Answer

Explanation: 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

View Answer

Explanation: The zeros of the quadratic polynomial cut the x-axis at two different points.

∴ b

^{2}– 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

View Answer

Explanation: 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

View Answer

Explanation: A number is called zero of polynomial when it satisfies the equation of the polynomial.

9. If a < 0, then the graph of ax^{2}+bx+c, has a downward opening.

a) True

b) False

View Answer

Explanation: The leading coefficient of the polynomial is less than zero, hence, it has downward opening. For example, the graph of -x

^{2}is

10. A polynomial is said to be linear, quadratic, cubic or biquadratic according to the degree of the polynomial.

a) False

b) True

View Answer

Explanation: 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) x^{2}+2x+5

b) √x+2x+4

c) x^{\(\frac {2}{3}\)}+10x

d) 5x+\(\frac {5}{x}\)

View Answer

Explanation: An expression in the form of (x)=a

_{0}+a

_{1}x+a

_{2}x

^{2}+…+a

_{n}x

^{n}, where a

_{n}≠0, is called a polynomial where a

_{1}, a

_{2}… a

_{n}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) (x^{2}+3)(x^{2}-3)

b) x^{2}-7

c) x^{7}+x^{6}+x^{5}

d) 5x-3

View Answer

Explanation: A biquadratic polynomial has highest power 4.

Hence, the polynomial with the highest power as 4 is x

^{4}-9 or (x

^{2}+3)(x

^{2}-3).

13. Which of the following is not a polynomial?

a) x^{2}+5x+10

b) √x+2x+4

c) x^{10}+10x

d) 5x+4

View Answer

Explanation: An expression in the form of (x)=a

_{0}+a

_{1}x+a

_{2}x

^{2}+…+a

_{n}x

^{n}, where a

_{n}≠0, is called a polynomial where a

_{1}, a

_{2}… a

_{n}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)

View Answer

Explanation: 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

View Answer

Explanation: 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.

**Sanfoundry Global Education & Learning Series – Mathematics – Class 10**.

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