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

$(x^{2} - 8 x + 1) (x^{2} - 8 x + 105)$

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

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$(x^{2} - 8 x + 21) (x^{2} - 8 x + 5)$

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$(x^{2} - 8 x + 8) (x^{2} - 8 x + 16)$

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Solution

The correct option is B
$(x^{2} - 8 x + 7) (x^{2} - 8 x + 15)$

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

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

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

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

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

Correct option : D

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Write x4−16x3+86x2−176x+105 as the product of two quadratic polynomial If one quadratic polynomial have roots 1 and 7.

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