Final answer:
To write a third-degree polynomial with rational coefficients using the given possible roots, we use the fact that if a is a root of a polynomial, then the polynomial has a factor of (x - a). The polynomial 2x³ + 7x² + x - 10 can be obtained by multiplying the factors (x - 1), (x + 1), (x - 2), (x + 2), (x - 5), (x + 5), (x - 10), and (x + 10).
Step-by-step explanation:
A polynomial is an algebraic expression that involves variables raised to non-negative integer powers and coefficients. The degree of a polynomial is the highest power of the variable in the expression.
To write a third-degree polynomial with rational coefficients using the given possible roots, we use the fact that if a is a root of a polynomial, then the polynomial has a factor of (x - a).
Since the given possible roots are ±1, ±2, ±5, ±10, we have the factors (x - 1), (x + 1), (x - 2), (x + 2), (x - 5), (x + 5), (x - 10), (x + 10). Multiplying these factors together, we get (x - 1)(x + 1)(x - 2)(x + 2)(x - 5)(x + 5)(x - 10)(x + 10). Expanding this expression gives us the polynomial 2x³ + 7x² + x - 10.
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