JEE PYQ: Quadratic Equation Question 5
Question 5 - 2021 (18 Mar Shift 2)
If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $P(x) = f(x^3) + xg(x^3)$ is divisible by $x^2 + x + 1$, then $P(1)$ is equal to
Show Answer
Answer: (0)
Solution
$P(x) = f(x^3) + xg(x^3)$
$P(1) = f(1) + g(1)$ …(1)
Now $P(x)$ is divisible by $x^2 + x + 1$
$\Rightarrow P(x) = Q(x)(x^2 + x + 1)$
$P(w) = 0 = P(w^2)$ where $w, w^2$ are non-real cube roots of unity.
$P(w) = f(w^3) + wg(w^3) = 0$
$f(1) + wg(1) = 0$ …(2)
$P(w^2) = f(w^6) + w^2g(w^6) = 0$
$f(1) + w^2g(1) = 0$ …(3)
$(2) + (3) \Rightarrow 2f(1) + (w + w^2)g(1) = 0$
$2f(1) = g(1)$ …(4)
$(2) - (3) \Rightarrow (w - w^2)g(1) = 0$
$g(1) = 0 = f(1)$ from (4)
From (1): $P(1) = f(1) + g(1) = 0$
BETA
