Mathematics Questions and Answers – Zeros and Coefficients of Polynomial – 1

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

1. The zeros of the polynomial 18x2-27x+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

Answer: a
Explanation: 18x2-27x+7=0
18x2-21x-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 \(\frac {1}{3}\).
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2. What will be the polynomial if its zeros are 3, -3, 9 and -9?
a) x4-80x2+729
b) x4-90x2+729
c) x4-90x2+79
d) x4-100x2+729
View Answer

Answer: b
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)
(x2-9) (x2-81) (By identity (x-a)(x+a)=x2-a2)
(x4-9x2-81x2+729)
x4-90x2+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) 2x2-20x+10
b) 2x2-x+5
c) 2x2-20x+5
d) x2-20x+5
View Answer

Answer: c
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)=x2-(α+β)x+αβ
f(x)=x2-10x+\(\frac {5}{2}\)
f(x)=2x2-20x+5
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4. If α and β are the zeros of x2+20x-80, then the value of α+β is _______
a) -15
b) -5
c) -10
d) -20
View Answer

Answer: d
Explanation: α and β are the zeros of x2+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 3x2-5x-15, then the value of αβ is _______
a) -5
b) -10
c) -15
d) -20
View Answer

Answer: a
Explanation: α and β are the zeros of 3x2-5x-15.
Product of zeros or αβ = \(\frac {constant \, term}{coefficient \, of \, x^2} = \frac {-15}{3}\) = -5
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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

Answer: b
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)
x2-5x-bx+5b
x2-(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 x2-10x+25.
Therefore, the other zero is 5.

7. The value of a and b, if the zeros of x2+(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

Answer: c
Explanation: The zeros of the polynomial are -5 and 9.
Hence, α=-5, β=9
The polynomial is x2+(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
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8. What will be the value of k, if one zero of x2+(k-3)x-16=0 is additive inverse of other?
a) 4
b) -4
c) -3
d) 3
View Answer

Answer: d
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 x2+(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 10x2+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

Answer: d
Explanation: \(\frac {1}{\alpha } + \frac {1}{\beta } = \frac {\alpha +\beta }{\alpha \beta }\)
α+β=\(\frac {-20}{10}\)=-2
αβ=\(\frac {-80}{10}\)=-8
∴ \(\frac {\alpha +\beta }{\alpha \beta } = \frac {-2}{-8} = \frac {1}{4}\)
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10. If α and β are the zeros of x2+35x-75, then _______
a) α+β<αβ
b) α+β>αβ
c) α+β=αβ
d) α+β≠αβ
View Answer

Answer: b
Explanation: The given polynomial is x2+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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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter