First slide

Theory of expressions

Question

Let $f (x) = x^{2} + bx + c$ where $b, c \in R$ . If f(x) is a factor of both $x^{4} + 6 x^{2} + 25 and 3 x^{4} + 4 x^{2} + 28 x + 5$ , then the least value of f(x) is

Moderate

A

2
B

3
C

5/2
D

4

Solution

$f (x) = x^{2} + bx + c$
f(x) is factor of both $x^{4} + 6 x^{2} + 25 and 3 x^{4} + 4 x^{2} + 28 x + 5$
f(x) will also be factor of
$\begin{array}{l} 3 (x^{4} + 6 x^{2} + 25) - (3 x^{4} + 4 x^{2} + 28 x + 5) = 14 (x^{2} - 2 x + 5) \\ ∴ f (x) = x^{2} - 2 x + 5 = {(x - 1)}^{2} + 4 \geq 4 \end{array}$

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