Find a polynomial of degree 3 with real coefficients that satisfies the given conditions.
-Zeros: $1 , - 2,3 ; f ( 2 ) = 20$
A) $- 5 x ^ { 3 } + 10 x ^ { 2 } + 25 x - 30$
B) $- 5 x ^ { 3 } - 20 x ^ { 2 } + 25 x - 30$
C) $5 x ^ { 3 } + 20 x ^ { 2 } - 25 x + 30$
D) $5 x ^ { 3 } - 10 x ^ { 2 } - 25 x + 30$

Question

Quizplus · Accepted Answer

Given the zeros $1, -2, 3$, the polynomial can be expressed as $k(x - 1)(x + 2)(x - 3)$, where $k$ is a constant. To find $k$, use the condition $f(2) = 20$: $k(2 - 1)(2 + 2)(2 - 3) = 20$, which simplifies to $k(1)(4)(-1) = 20$, so $k = -5$. Thus, the polynomial is $-5(x - 1)(x + 2)(x - 3)$, which expands to $-5x^3 + 10x^2 + 25x - 30$.

Find a Polynomial of Degree 3 with Real Coefficients That 1,−2,3;f(2)=201 , - 2,3 ; f ( 2 ) = 201,−2,3;f(2)=20

Find a Polynomial of Degree 3 with Real Coefficients That $1 , - 2,3 ; f ( 2 ) = 20$