Math, asked by swetanks2007, 11 hours ago

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Answers

Answered by tennetiraj86
4

Step-by-step explanation:

Given :-

(x+2a) is a factor of x⁴-2ax³+2x+2a+3

To find :-

Find the value of a ?

Solution:-

Given bi-quadratic polynomial is x⁴-2ax³+2x+2a+3

Let P(x) = x⁴-2ax³+2x+2a+3

Given factor = (x+2a)

We know that

Factor Theorem:-

Let P(x) be a polynomial of the degree greater than or equal to 1 and x-a is another linear polynomial if x-a is a factor of P (x) then P(a) = 0 vice-versa.

Given that

(x+2a) is a factor of P(x)

=> P(-2a) = 0

Since x+2a = 0 => x = -2a

Put x = -2a then P(-2)

=> (-2a)⁴-2a(-2a)³+2(-2a)+2a+3 = 0

=> 16a⁴-2a(-8a³)-4a+2a+3 = 0

=> 16a⁴+16a⁴-4a+2a+3 = 0

=> (16+16)a⁴+(-4+2)a +3 = 0

=> 32a⁴-2a+3 = 0

There is no real value for a for the given Polynomial.

Correction:-

If the given polynomial is x⁴+2ax³+2x+2a+3 then we get real value of a

Now P(-2a) = 0

=> (-2a)⁴+2a(-2a)³+2(-2a)+2a+3 = 0

=> 16a⁴-16a⁴-4a+2a+3 = 0

=> 0-2a+3 = 0

=> -2a +3 = 0

=> -2a = -3

=> 2a = 3

=> a = 3/2

The value of a = 3/2

Answer :-

The value of a for the given problem is 3/2

Check:-

if a = 3/2 then x+2a = x+2(3/2) = x+3

If x+3 is a factor of P(x) then P(-3) = 0

=> (-3)⁴+2(3/2)(-3)³+2(-3)+2(3/2)+3

=> 81-81-6+3+3

=> 81-81-6+6

=> 0+0

=> 0

So, x-3 is a factor of P(x) when a = 3/2

Used formulae:-

Factor Theorem:-

Let P(x) be a polynomial of the degree greater than or equal to 1 and x-a is another linear polynomial if x-a is a factor of P (x) then P(a) = 0 vice-versa.

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