Math Olympiad Test: Polynomials - 1 - Class 10 MCQ

Given a and b are the zeros of ax² + bx+ c
∴ 

P(x) = x² - 8x + k
If α, and β are its zeros
then α + β = 8, αβ = k
and α² + β² = 40
⇒ (α + β)² - 2αβ = 8² - 2k
⇒ 40 = 64 - 2k
⇒ 2k = 24
⇒ k = 12

Here, (4x² + 3x - 2) 8x⁴ + 14x³ - 2x² + 7x - 8 (2x² + 2x - 1)



∴ 14x - 10 must be subtracted.

Given a + b+ γ = 2 = ⇒ b = -2a
αβ + βγ + γα = -7 = ⇒ c = -7a
αβγ = -14 = ⇒ d = 14a

b : c : d = -2 : -7 : 14
Cubic polynomial is
k(x³ - 2x² - 7x + 14)

If α and β are two zeros then αβ = 3 Let third root be γ.

The given polynomial is
2 x³ + x² − 13x + 6
α,β,γ be its zeros
∴ 

Here
(x² + 2 x — 3)4 x⁴ + 2 x³ — 2 x² + x — l(4 x² — 6 x + 22

Clearly -61x + 65 must be added.

Required polynomial
= P(x ) = x³ - (α + β + γ)x² + (αβ + βγ + γα)x
- αβγ
Thus, the cubic polynomial is
p(x) = x³ − (4)x² + (1)x − (−6)
= x³ − 4x²+ x + 6

Given data 
Sum of zeroes = −5
Product of zeroes = -12
Equation of quadratic polynomial
x² − (sum of zeroes) x + product of zeroes = 0
x² − (−5)x + (-12) = 0
x² + 5x - 12 = 0

We have x² - 2x - 3
∴