Question

If (ax + b)is a factor of x^2 +bx +a .find value of b in terms of a

Answer 1

Step-by-step explanation:

Given :-

(ax + b) is a factor of x² +bx +a

To find :-

Find value of b in terms of a ?

Solution :-

Given quardratic polynomial is x² +bx +a

Let P(x) = x² +bx +a

Given factor = ax+b

We know that

By Factor Theorem

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

Given that

(ax + b) is a factor of x²+bx +a

=> P(-b/a) = 0

( Since ax+b = 0

=> ax = -b

=>x = - b/a )

Now,

Put x = -b/a then

P(-b/a) = 0

=> (-b/a)² +b (-b/a) + a = 0

=> (b²/a²) + (-b²/a) + a = 0

=>(b²/a²) -(b²/a) + a = 0

LCM of a and a² = a²

=>(b² -ab²+a³)/a² = 0

=> b²-ab²+a³ = 0×a²

=> b²-ab²+a³ = 0

=> b²(1-a) + a³ = 0

=> b²(1-a) = -a³

=>b² = -a³/(1-a)

=>b²= -a³/-(a-1)

=>b² = a³/(a-1)

=> b =√[a³/(a-1)]

Therefore, b =√[a³/(a-1)]

Answer:-

The value of b in terms of a for the given problem is √[a³/(a-1)]

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)=p vice-versa.