G Intercepts 2B7B04

1. **State the problem:** We are given the function $$g(n) = 2n^4 - 16n^3 + 14n^2$$ and need to find its $g$-intercept and $n$-intercepts. 2. **Find the $g$-intercept:** The $g$-intercept is the value of $g(n)$ when $n=0$. $$g(0) = 2(0)^4 - 16(0)^3 + 14(0)^2 = 0$$ So, the $g$-intercept is at the point $(0,0)$. 3. **Find the $n$-intercepts:** The $n$-intercepts occur where $g(n) = 0$. Set the function equal to zero: $$2n^4 - 16n^3 + 14n^2 = 0$$ 4. **Factor the equation:** First, factor out the greatest common factor $2n^2$: $$2n^2(n^2 - 8n + 7) = 0$$ 5. **Solve each factor:** - From $2n^2 = 0$, we get: $$n^2 = 0 \implies n = 0$$ - From $n^2 - 8n + 7 = 0$, solve the quadratic equation: $$n = \frac{8 \pm \sqrt{(-8)^2 - 4 \times 1 \times 7}}{2} = \frac{8 \pm \sqrt{64 - 28}}{2} = \frac{8 \pm \sqrt{36}}{2}$$ $$n = \frac{8 \pm 6}{2}$$ So, $$n = \frac{8 + 6}{2} = 7 \quad \text{or} \quad n = \frac{8 - 6}{2} = 1$$ 6. **List all $n$-intercepts:** $$n = 0, 1, 7$$ 7. **Summary:** - The $g$-intercept is at $(0,0)$. - The $n$-intercepts are at $n=0$, $n=1$, and $n=7$. These are the points where the graph crosses the $n$-axis.