1. **State the problem:** We analyze polynomial $P(x)$ with given properties: constant 4th finite differences equal to 48, three distinct x-intercepts at $x=-3$, $x=-1$, and $x=2$, a local minimum at $(-3,0)$, and y-intercept at $(0,-36)$. We check which statements about $P(x)$ are true. 2. **Degree of polynomial:** Constant 4th finite differences imply $P(x)$ is degree 4. 3. **Form of polynomial:** Since $P(x)$ has roots at $-3$, $-1$, and $2$, and $-3$ is a local minimum (touching x-axis but not crossing), root at $-3$ has multiplicity 2. So, $$P(x) = a(x+3)^2(x+1)(x-2)$$ 4. **Find leading coefficient $a$ using y-intercept:** At $x=0$, $P(0) = -36$, $$-36 = a(3)^2(1)(-2) = a \times 9 \times (-2) = -18a$$ So, $$a = \frac{-36}{-18} = 2$$ 5. **Check 4th finite differences:** For degree 4 polynomial, $$\text{4th finite difference} = 4! \times a = 24 \times 2 = 48$$ Matches given data. 6. **Check behavior as $x \to \infty$:** Leading term is $2x^4$, which goes to $+\infty$, so $$\lim_{x \to \infty} P(x) = +\infty$$ 7. **Number of turning points:** Degree 4 polynomial can have up to 3 turning points. Here, with roots and multiplicities, exactly 3 turning points exist. 8. **Absolute minimum:** Since $P(x) \to +\infty$ as $x \to \pm \infty$ and has a local minimum at $(-3,0)$, this local minimum is also the absolute minimum. 9. **3rd finite differences:** 3rd finite differences are not constant but change; since 4th finite differences are constant positive, 3rd finite differences form an increasing sequence, so 3rd finite differences are not constant. 10. **Range:** Since polynomial has absolute minimum at 0 and goes to $+\infty$, range is $[0, \infty)$, not all real numbers. 11. **Point symmetry:** Polynomial is degree 4 with mixed terms, no point symmetry about origin. **Summary of true statements:** - The degree of the polynomial is 4. - The function possesses an absolute minimum value. - The function has exactly three turning points. - The x-intercept at $x = -3$ has multiplicity 2. - The y-intercept of the function is located at $(0, -36)$. - The leading coefficient of the function is 2. **False statements:** - The 3rd finite differences are a constant positive value. - As $x \to \infty$, $P(x) \to -\infty$. - The range is all real numbers. - The function possesses point symmetry about the origin.