If one root of 10x^3  x^2  278x + 165 = 0 is 5, then the product of the remaining two roots is C 33/10.
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:

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

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

Perform polynomial division: Divide (10x^3  x^2  278x + 165) by ((x  5)):
[ \begin{array}{rrrrr} 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 ]

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} ]

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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