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If both x - 2 and x - 1/2 are factors of px² + 5x + r, show that p = r.

Solution:

Given, the polynomial is px² + 5x + r

x - 2 and x - 1/2 are factors of the polynomial.

We have to show that p = r.

Let f(x) = px² + 5x + r

Now, g(x) = x - 2

g(x) = 0

x - 2 = 0

x = 2

Put x = 2 in p(x),

p(2) = p(2)² + 5(2) + r

= 4p + 10 + r

At x = 2, p(x) = 0

So, 4p + 10 + r = 0 —-------------- (1)

Now, h(x) = x - 1/2

Put x = 1/2 in p(x),

p(1/2) = p(1/2)² + 5(1/2) + r

= p/4 + 5/2 + r

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

= (p + 10 + 4r)/4

At x = 1/2, p(x) = 0

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

p + 10 + 4r = 0 —------------ (2)

On comparing (1) and (2),

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

By grouping,

4p - p + 10 - 10 = 4r - r

3p = 3r

p = r

Therefore, it is proved that p = r.

✦ Try This: If x(square) - 1 is a factor of ax(cube) + bx(square) + cx + d,show that a+c=0.

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 2

NCERT Exemplar Class 9 Maths Exercise 2.4 Problem 3

If both x - 2 and x - 1/2 are factors of px² + 5x + r, show that p = r

Summary:

A polynomial is a type of expression in which the exponents of all variables should be a whole number. If both x - 2 and x - 1/2 are factors of px² + 5x + r, it is shown that p = r

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