Question

If ${x^2} - 3x + 2$ be a factor of ${x^4} - p{x^2} + q$ , then $(p,q) = $
A. $(3,4)$
B. $(4,5)$
C. $(4,3)$
B. $(5,4)$

Answer 1

Hint: We are provided here two expressions. One expression is quadratic and a factor of the other expression. So, we can find the roots of the first expression using the Sridharacharya formula. And as we know, if one expression is a factor of another then, the roots of the equation will also satisfy the other equation. We have to use this concept only and further we will get two simultaneous equations. On solving, we will get the final result.

Formula Used: The roots of a quadratic equation are found using the quadratic formula.
$\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$

Complete step-by-step solution:
We have been given two expressions as follows:
${x^2} - 3x + 2$ --(1.1)
And
${x^4} - p{x^2} + q$ --(1.2)
As given, the equation (1.1) is a factor of the equation (1.2). So, first we will find the roots of the equation (1.1).
To find the roots, compare equation (1.1) with the standard quadratic equation $a{x^2} + bx + c = 0$ .
We get $a = 1$ , $b = - 3$ and $c = 2$ .
Substituting these values in the quadratic formula $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ to get the value for $x$
$\therefore $ $x = \dfrac{{ - ( - 3) \pm \sqrt {{{( - 3)}^2} - 4 \times 1 \times 2} }}{{2 \times 1}}$ .
$ \Rightarrow x = \dfrac{{3 \pm \sqrt 1 }}{2}$
So, we have
$ \Rightarrow x = 2,1$
Now, we substitute these values of $x$ in the equation (1.2), we will get following two expressions:
${(2)^4} - {(2)^2}p + q$ and $p - q - 1$ .
To get the value of the unknowns we equalize these to zero, to express them as equations.
$4p - q - 16 = 0$ and $p - q - 1 = 0$
Solving these equations simultaneously, we get the result that
$(p,q) = (5,4)$ .
Hence, the correct option is D.

Note: It should be kept in mind that the first equation is a factor of the other equation. So, we will only find the roots of the first quadratics. If we find the roots of the second expression then, it will not satisfy the first expression as the second expression is not a factor of the first one. Taking that under consideration, we have to satisfy the second expression only.