This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Zeros and Coefficients of Polynomial – 1”.

1. The zeros of the polynomial 18x^{2}-27x+7 are ___________

a) \(\frac {7}{6}, \frac {1}{3}\)

b) \(\frac {-7}{6}, \frac {1}{3}\)

c) \(\frac {7}{6}, \frac {-1}{3}\)

d) \(\frac {7}{3}, \frac {1}{3}\)

View Answer

Explanation: 18x

^{2}-27x+7=0

18x

^{2}-21x-6x+7=0

3x(6x-7)-1(6x-7)=0

(6x-7)(3x-1)=0

x=\(\frac {7}{6}, \frac {1}{3}\)

The zeros are \(\frac {7}{6}\) and \(\frac {1}{3}\).

2. What will be the polynomial if its zeros are 3, -3, 9 and -9?

a) x^{4}-80x^{2}+729

b) x^{4}-90x^{2}+729

c) x^{4}-90x^{2}+79

d) x^{4}-100x^{2}+729

View Answer

Explanation: The zeros of the polynomial are 3, -3, 9 and -9.

Then, (x-3), (x+3), (x-9) and (x+9) are the factors of the polynomial.

Multiplying the factors, we have

(x-3) (x+3) (x-9) (x+9)

(x

^{2}-9) (x

^{2}-81) (By identity (x-a)(x+a)=x

^{2}-a

^{2})

(x

^{4}-9x

^{2}-81x

^{2}+729)

x

^{4}-90x

^{2}+729

3. The sum and product of zeros of a quadratic polynomial are 10 and \(\frac {5}{2}\) respectively. What will be the quadratic polynomial?

a) 2x^{2}-20x+10

b) 2x^{2}-x+5

c) 2x^{2}-20x+5

d) x^{2}-20x+5

View Answer

Explanation: The sum of the polynomial is 10, that is, α+β = 10

The product of the polynomial is \(\frac {5}{2}\) i.e. αβ = \(\frac {5}{2}\)

∴ f(x)=x

^{2}-(α+β)x+αβ

f(x)=x

^{2}-10x+\(\frac {5}{2}\)

f(x)=2x

^{2}-20x+5

4. If α and β are the zeros of x^{2}+20x-80, then the value of α+β is _______

a) -15

b) -5

c) -10

d) -20

View Answer

Explanation: α and β are the zeros of x

^{2}+20x-80.

Sum of zeros or α+β = \(\frac {-coefficient \, of \, x}{coefficient \, of \, x^2} = \frac {-20}{1}\) = -20

5. If α and β are the zeros of 3x^{2}-5x-15, then the value of αβ is _______

a) -5

b) -10

c) -15

d) -20

View Answer

Explanation: α and β are the zeros of 3x

^{2}-5x-15.

Product of zeros or αβ = \(\frac {constant \, term}{coefficient \, of \, x^2} = \frac {-15}{3}\) = -5

6. What will be the value of other zero, if one zero of the quadratic polynomial is 5 and the sum of the zeros is 10?

a) 10

b) 5

c) -5

d) -10

View Answer

Explanation: One zero of the quadratic polynomial is 5. ∴ the factor of the polynomial is (x-5)

Let us assume the other zero to be b. ∴ the other factor of the polynomial is (x-b)

Multiplying the factors, we have (x-5)(x-b)

x

^{2}-5x-bx+5b

x

^{2}-(5+b)x+5b

The sum of zeros is 10.

∴ \(\frac {-coefficient \, of \, x}{coefficient \, of \, x^2}\)=10

\(\frac {-(-5-b)}{1}\)=10

5+b=10

b=5

The equation becomes x

^{2}-10x+25.

Therefore, the other zero is 5.

7. The value of a and b, if the zeros of x^{2}+(a+5)x-(b-4) are -5 and 9 will be _________

a) 47, -5

b) -5, 47

c) -9, 49

d) -4, 45

View Answer

Explanation: The zeros of the polynomial are -5 and 9.

Hence, α=-5, β=9

The polynomial is x

^{2}+(a+5)x-(b-4).

Sum of zeros or α+β=-5+9 = \(\frac {-coefficient \, of \, x}{coefficient \, of \, x^2} = \frac {a+5}{1}\)

-4=a+5

a = -9

Product of zeros or αβ = -45 = \(\frac {constant \, term}{coefficient \, of \, x^2} = \frac {-(b-4)}{1}\)

-45=-b+4

b=49

8. What will be the value of k, if one zero of x^{2}+(k-3)x-16=0 is additive inverse of other?

a) 4

b) -4

c) -3

d) 3

View Answer

Explanation: Since, one zero of the polynomial is the additive inverse of the other.

Hence, the sum of roots will be zero.

The polynomial is x

^{2}+(k-3)x-16=0

Sum of zeros or α+β=\(\frac {-coefficient \, of \, x}{coefficient \, of \, x^2} = \frac {k-3}{1}\)=0

k-3=0

k=3

9. If α and β are the zeros of 10x^{2}+20x-80, then the value of \(\frac {1}{\alpha } + \frac {1}{\beta }\) is _______

a) \(\frac {5}{4}\)

b) \(\frac {1}{5}\)

c) \(\frac {3}{4}\)

d) \(\frac {1}{4}\)

View Answer

Explanation: \(\frac {1}{\alpha } + \frac {1}{\beta } = \frac {\alpha +\beta }{\alpha \beta }\)

α+β=\(\frac {-20}{10}\)=-2

αβ=\(\frac {-80}{10}\)=-8

∴ \(\frac {\alpha +\beta }{\alpha \beta } = \frac {-2}{-8} = \frac {1}{4}\)

10. If α and β are the zeros of x^{2}+35x-75, then _______

a) α+β<αβ

b) α+β>αβ

c) α+β=αβ

d) α+β≠αβ

View Answer

Explanation: The given polynomial is x

^{2}+35x-75.

The sum of zeros, α + β = \(\frac {-coefficient \, of \, x}{coefficient \, of \, x^2} = \frac {-35}{1}\) = -35

The product of zeros, αβ = \(\frac {constant \, term}{coefficient \, of \, x^2}\) = -75

Clearly, sum of zeros is greater than product of zeros.

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