Question

Let P(x) = x^{4} – 5x^{3} + 9x^{2} – 7x + 2 and Q(x) = x^{2} + x + a. Find the number of possible values of
‘a’ for which there exists a common factor between P(x) and Q(x)
please answer the question

Answer 1

Answer:

Consider p(x)

Factors of 2 (the constant term) are 1, -1, 2, -2

p(1) = 1 - 5 + 9 - 7 + 2 = 0

Hence, (x - 1) is a factor of p(x)

We rewrite p(x) as

x⁴ - x³ - 4x³ + 4x² + 5x² - 5x - 2x + 2

So we get

x³(x-1) - 4x²(x-1) + 5x(x-1) - 2(x-1)

= (x - 1)(x³ - 4x² + 5x -1)

Now we consider x³ - 4x² + 5x - 1 = f(x) (say)

Factors of -1 (the constant term) are 1 and -1

f(1) = 1 - 4 + 5 - 1 = 1 not equal to 0

f(-1) = -1 - 4 - 5 - 1 = -11 not equal to zero.

Hence, f(x) does not have a factor

Hence the factors of p(x) are x - 1 and x³ - 4x² + 5x - 1 (=f(x))

p(x) and q(x) cannot have a common factor f(x) since f(x) is of higher degree than q(x).

So the only common factor of p(x) and q(x) is x - 1

Now, Consider q(x)

When divided by (x - 1), q(x) will leave the remainder q(1)

q(1) = 1² + 1 + a = 2 + a

We want x - 1 to be a factor of q(x), so the remainder is zero

Thus, 2 + a = 0 => a = -2

Hence, the required value of a is -2