Write the function as a product of linear and irreducible quadratic factors, all with real coefficients.
-$$f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } + 3 x - 10$$
A) $( x - 2 ) \left( 2 x ^ { 2 } + x + - 5 \right)$
B) $( x - 2 ) \left( 2 x ^ { 2 } + x + 5 \right)$
C) $( x + 2 ) \left( 2 x ^ { 2 } + x + - 5 \right)$
D) $( x + 2 ) \left( 2 x ^ { 2 } + x + 5 \right)$

Question

Quizplus · Accepted Answer

To factor the given cubic polynomial $2x^3 - 3x^2 + 3x - 10$, we first look for rational roots using the Rational Root Theorem. The possible rational roots are factors of the constant term (-10) divided by factors of the leading coefficient (2), which are $\pm1, \pm2, \pm5, \pm10, \pm\frac{1}{2}, \pm\frac{5}{2}$. Testing these, we find that $x = 2$ is a root because substituting $x = 2$ into the polynomial gives $0$. Therefore, $x - 2$ is a factor. Dividing the polynomial by $x - 2$ gives the quadratic factor $2x^2 + x + 5$, which cannot be factored further into real linear factors because its discriminant $b^2 - 4ac = 1^2 - 4(2)(5) = 1 - 40 = -39$ is negative, indicating it has no real roots. Thus, the factorization is $(x - 2)(2x^2 + x + 5)$.

Write the Function as a Product of Linear and Irreducible f(x)=2x3−3x2+3x−10f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } + 3 x - 10f(x)=2x3−3x2+3x−10

Write the Function as a Product of Linear and Irreducible $f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } + 3 x - 10$