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

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Solution

The correct option is D

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

If 1 and 7 are the roots of the given equation. (x - 1)(x - 7) are the factors of $x^{4} - 16 x^{3} + 86 x^{2} - 176 x + 105 = 0$ .

Divide $x^{4} - 16 x^{3} + 86 x^{2} - 176 x + 105 b y x^{2} - 8 x + 7$ and find the quotient

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

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

Correct option : D

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