1. **Problem Statement:** Graph the polynomial function $f(x) = x^3 + x^2 - 4x - 4$ given that $(x+1)$ is a zero. 2. **Use the Factor Theorem:** Since $(x+1)$ is a zero, $f(-1) = 0$. This means $(x+1)$ is a factor of $f(x)$. 3. **Divide $f(x)$ by $(x+1)$ to find the other factors:** Use polynomial division or synthetic division: $$\frac{x^3 + x^2 - 4x - 4}{x+1}$$ Perform synthetic division: - Coefficients: 1 (for $x^3$), 1 (for $x^2$), -4 (for $x$), -4 (constant) - Divide by root $-1$: Carry down 1. Multiply $1 \times (-1) = -1$, add to 1: $1 + (-1) = 0$. Multiply $0 \times (-1) = 0$, add to -4: $-4 + 0 = -4$. Multiply $-4 \times (-1) = 4$, add to -4: $-4 + 4 = 0$. Remainder is 0, quotient is $x^2 + 0x - 4 = x^2 - 4$. 4. **Factor the quotient:** $$x^2 - 4 = (x - 2)(x + 2)$$ 5. **Write the full factorization:** $$f(x) = (x + 1)(x - 2)(x + 2)$$ 6. **Find zeros:** Set each factor to zero: - $x + 1 = 0 \Rightarrow x = -1$ - $x - 2 = 0 \Rightarrow x = 2$ - $x + 2 = 0 \Rightarrow x = -2$ 7. **Graph behavior:** - The polynomial is cubic with leading coefficient positive, so it falls to the left and rises to the right. - Zeros at $-2, -1, 2$. --- 1. **Problem Statement:** Graph $g(x) = x^3 + 4x^2 + 4x$ and factor to find zeros. 2. **Factor $g(x)$:** Factor out the greatest common factor (GCF): $$g(x) = x(x^2 + 4x + 4)$$ 3. **Factor the quadratic:** $$x^2 + 4x + 4 = (x + 2)^2$$ 4. **Full factorization:** $$g(x) = x(x + 2)^2$$ 5. **Find zeros:** Set each factor to zero: - $x = 0$ - $x + 2 = 0 \Rightarrow x = -2$ (with multiplicity 2) 6. **Graph behavior:** - Cubic with leading coefficient positive. - Zero at $x=0$ (crosses x-axis). - Zero at $x=-2$ (touches x-axis and turns around due to multiplicity 2). --- **Summary:** - $f(x) = (x + 1)(x - 2)(x + 2) = x^3 + x^2 - 4x - 4$ - $g(x) = x(x + 2)^2 = x^3 + 4x^2 + 4x$ These are the standard form equations and their factorizations.