Question

answer the question with step by step explanation​

Answer 1

Step-by-step explanation:

Given :-

(x-2) and (x -1/2) are factors of px²+5x+r .

To find :-

Prove that p = r .

Solution :-

Given quadratic polynomial is px²+5x+r

Let g(x) = px²+5x+r

Given factors are (x-2) and (x-1/2)

We know that : Factor Theorem

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

If (x-2) is a factor of g(x) then g(2) = 0

=> g(2) = p(2)²+5(2)+r = 0

=> p(4)+10+r = 0

=> 4p +r +10 =0 -------------(1)

and

If (x- 1/2) is a factor of g(x) then g(1/2) = 0

=> g(1/2) = p(1/2)²+5(1/2)+r = 0

=> p(1/4)++(5/2)+r = 0

=> (p/4)+(5/2)+r = 0

LCM of 4 and 2 is 4

=> (p+10+4r)/4 = 0

=> p+4r+10 = 0×4

=> p+4r+10 = 0-------------(2)

Given that

(x-2) and (x-1/2) both are factors of the given polynomial then

g(2) = g(1/2)

=>4p+r+10 = p+4r+10

=> 4p+r+10-p-4r-10 = 0

=> (4p-p)+(r-4r)+(10-10) = 0

=> (3p)+(-3r)+0 = 0

=> 3p-3r = 0

> 3p = 3r

On cancelling 3 both sides then

=> p = r

Hence, Proved.

Answer :-

If (x-2) and (x -1/2) are factors of px²+5x+r then

p = r

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.