Question

Write $x^4-16x^3+86x^2-176x+105$ as the product of two quadratic polynomial If one quadratic polynomial have roots $1$ and $7$.

• A. $(x^2-8x+7)(x^2-8x+15)$
• B. $(x^2-8x+1)(x^2-8x+105)$
• C. $(x^2-8x+21)(x^2-8x+5)$
• D. $(x^2-8x+8)(x^2-8x+16)$

Answer 1

The correct option is A

$(x^2-8x+7)(x^2-8x+15)$

Quadratic polynomial with roots 1 and 7 = $(x-1)(x-7)$
= $x^2-8x+7$

If 1 and 7 are the roots of the given equation.
$\Rightarrow (x-1),(x-7)$ are the factors of $x^4-16x^3+86x^2-176x+105$.

Divide $x^4-16x^3+86x^2-176x+105$ by $x^2-8x+7$

$\begin{array}{cc}& x^2+8x+15\\ x^2-8x+7& x^4-16x^3+86x^2-176x+105\\ & \_x^4\_+-8x^3\_-+7x^2\\ & -8x^3+79x^2-176x+105\\ & {}_{+}-8x^3{}_{-}+64x^2{}_{+}-56x\\ & 15x^2-120x+105\\ & \_-15x^2+\_+120x{}_{-}+105\\ & 0\end{array}$
Quotient is $x^2-8x+15$

$\Rightarrow x^4-16x^3+86x^2-176x+105=(x^2-8x+7)\times (x^2-8x+15)$

Correct option : D