Polynomial Intercepts 456651

1. **State the problem:** We are given the polynomial function $$P(x) = (x - 2)^2 (x - 5)$$ and need to find its y-intercept, x-intercepts, and end behavior as $x \to \infty$ and $x \to -\infty$. 2. **Find the y-intercept:** The y-intercept occurs when $x=0$. Substitute $x=0$ into the function: $$P(0) = (0 - 2)^2 (0 - 5) = ( -2)^2 \times (-5) = 4 \times (-5) = -20$$ So, the y-intercept is at $(0, -20)$. 3. **Find the x-intercepts:** The x-intercepts occur where $P(x) = 0$. Set the function equal to zero: $$(x - 2)^2 (x - 5) = 0$$ This product is zero if either factor is zero: - $(x - 2)^2 = 0 \implies x = 2$ - $(x - 5) = 0 \implies x = 5$ So, the x-intercepts are at $x=2$ (with multiplicity 2) and $x=5$. 4. **Determine end behavior:** The degree of the polynomial is $3$ (since $(x-2)^2$ is degree 2 and $(x-5)$ is degree 1, total degree $2+1=3$). Leading term expansion: $$(x - 2)^2 (x - 5) = (x^2 - 4x + 4)(x - 5)$$ Multiply out: $$x^3 - 5x^2 - 4x^2 + 20x + 4x - 20 = x^3 - 9x^2 + 24x - 20$$ The leading term is $x^3$ which dominates for large $|x|$. - As $x \to \infty$, since the leading term is $x^3$, which grows positively, $P(x) \to \infty$. - As $x \to -\infty$, since $x^3$ is negative for negative $x$, $P(x) \to -\infty$. **Final answers:** - y-intercept: $(0, -20)$ - x-intercepts: $x=2$ (double root), $x=5$ - As $x \to \infty$, $P(x) \to \infty$ - As $x \to -\infty$, $P(x) \to -\infty$