Question

If one root of $10x^{3} - x^{2} - 278x + 165 = 0$ is 5, then the product of the remaining two roots is C $-\frac{33}{10}$.

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Answer

To determine the product of the remaining two roots of the polynomial (10x^3 - x^2 - 278x + 165 = 0), given that one root is (5), follow these steps:

  1. Given polynomial equation: [ 10x^3 - x^2 - 278x + 165 = 0 ]

  2. Confirm that (5) is a root: Since (5) is a root, the polynomial is divisible by ((x - 5)).

  3. Perform polynomial division: Divide (10x^3 - x^2 - 278x + 165) by ((x - 5)):

    [ \begin{array}{r|rrrr} 10x^2 & 49x & -33 & 0 \ \hline x - 5 & 10x^3 & -x^2 & -278x & +165 \ & 10x^3 & -50x^2 & +245x & -165 \ \hline & 0 & 49x^2 & -33x & 0 \ \end{array} ]

    After performing this division, we obtain the quadratic factor: [ 10x^2 + 49x - 33 ]

  4. Finding the remaining roots: We need to solve for the roots of the quadratic (10x^2 + 49x - 33): [ \begin{aligned} 10x^2 + 49x - 33 &= 0 \ \implies (5x - 3)(2x + 11) &= 0 \end{aligned} ]

    Solving these factors: [ \begin{aligned} 5x - 3 &= 0 \implies x = \frac{3}{5} \ 2x + 11 &= 0 \implies x = -\frac{11}{2} \end{aligned} ]

  5. Product of the remaining two roots: The two remaining roots are ( \frac{3}{5} ) and ( -\frac{11}{2} ).

    Their product is: [ \frac{3}{5} \times -\frac{11}{2} = -\frac{33}{10} ]

    Therefore, the correct answer is ( \boxed{-\frac{33}{10}} ).

The product of the remaining two roots is (\boxed{-\frac{33}{10}}), which corresponds to option C.


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