Solve the equation $x^{4} - 16 x^{3} + 86 x^{2} - 176 x + 105 = 0$ . If two roots being 1 and 7, Find the sum of the square of other two roots.

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Solution

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 $x^{2}$ - 176x + 105 = 0. Divide $x^{4}$ - 16 $x^{3}$ + 86 $x^{2}$ - 176x + 105 by $x^{2}$ - 8x + 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 $x^{2}$ - 8x + 15 = 0 (x - 3) (x - 5) = 0 x = 3, 5 Sum of the square of the square of other two roots = $3^{2}$ + $5^{2}$ = 9 + 25 = 34.

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Q. Solve the equations:

x^{4} - 16 x^{3} + 86 x^{2} - 176 x + 105 = 0

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1

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7

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are the roots of the equation

x^{4} - 16 x^{3} + 86 x^{2} - 176 x + 105 = 0

, then match the elements of List I with elements of List II:

List I	List II
I) $\sum α$	a) $86$
II) $\sum α β$	b) $105$
III) $\sum α β γ$	c) $16$
IV) $α β γ δ$	d) $176$

Correct order for I, II, III, IV is:

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Solve the equation x4−16x3+86x2−176x+105=0 . If two roots being 1 and 7, Find the sum of the square of other two roots. __

Solve the equation $x^{4} - 16 x^{3} + 86 x^{2} - 176 x + 105 = 0$ . If two roots being 1 and 7, Find the sum of the square of other two roots.

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