Answer:
- cannot be factored further using integer coefficients
- can be factored to linear factors using irrational constants
Explanation:
You want to know if the binomial 2x³ -10x can be factored further than 2x(x² -5) and how you can tell.
Factored form
Usually, when we say "factor p(x)", where p(x) is a polynomial, we mean we want it factored to factors that have integer coefficients. In many cases, the result will be a product of linear and/or quadratic factors. Here, ...
2x³ -10x = 2x(x² -5) . . . . . is factored to integers
When we want the polynomial "factored completely", or "factored to linear factors" the restriction that coefficients be integers or real numbers is removed. The difference of squares can be factored further, but irrational values are required:
= 2x(x -√5)(x +√5) . . . . . factored completely to linear factors
You can tell if further factoring is possible by looking at the roots of the non-linear factors in question. If those roots are rational, you can continue factoring. If they are irrational or complex, then the problem statement will tell you whether further factoring is desired.
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Additional comment
Of course, you need to be familiar with methods of factoring and of determining the possible rational roots of a polynomial. (A good graphing calculator can help.)