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Selecting Consecutive Terms in A.P

If the equati...

Question

If the equation $2 x^{4} + 7 x^{3} + a x + b = 0$ has four roots (All real roots) then find the value of a and b. Given that (x - 3) and (x - 1) may exactly divide the above given expression.

a = 12, b = 5

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a = 11, b = - 6

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a = 6, b = - 11

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a = 12, b = - 5

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Solution

The correct option is B
a = 11, b = - 6

(x - 1) and (x - 3) exactly divide the equation

Put x = 1

2 - 7 + a + b = 0

a + b = 5 _ (1)

Put x = 3

162 -7 × 27 + 39 + b = 0

3a + b = - 27 _ (2)

Solving equation 1 and 2

we get,

a = 11, b = - 6.

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Similar questions

If the equation $x^{4} - 4 x^{3} + a x^{2} + b x + 1 = 0$ has four roots. All of them are positive real roots then value of a and b are given.

A = {x | x is an integer root of the equation x^{5} - 6 x^{4} + 11 x^{3} - 6 x^{2} = 0}

B = {x | x is a real root of the equation a x^{5} + 2 a x^{3} + 2 b x^{2} + b = 0, a, b \in R such that the given equation have maximum number of real roots.}

x^{2} + 2 (a + b + c) x + 3 λ (a b + b c + a c) = 0

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λ < \frac{4}{3}

\frac{11}{3} < λ < \frac{17}{3}

\frac{11}{6} < λ < \frac{15}{4}

λ \geq 1

x^{2} - 10 c x - 11 d = 0

x^{2} - 10 a x - 11 b = 0

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a

b

2 x^{4} - 7 x^{3} + a x + b

x - 3

a, b

x^{3} - a x^{2} + b x - 6 = 0

b

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