1. **State the problem:** We are given two polynomial functions: - $y = 2x(x + 3)(x - 2)(x - 4)$ - $y = (x^2 + 3)(x^2 - 4)$ We need to find for each: - Degree - Zeros and their types - Y-intercept - End behavior 2. **Analyze the first polynomial:** - The function is $y = 2x(x + 3)(x - 2)(x - 4)$. - This is a product of four linear factors multiplied by 2. **Degree:** Each factor is degree 1, so total degree is $1 + 1 + 1 + 1 = 4$. **Zeros and types:** Set each factor equal to zero: - $2x = 0 \Rightarrow x = 0$ - $x + 3 = 0 \Rightarrow x = -3$ - $x - 2 = 0 \Rightarrow x = 2$ - $x - 4 = 0 \Rightarrow x = 4$ All zeros are distinct and linear, so all are simple zeros (multiplicity 1). **Y-intercept:** Evaluate $y$ at $x=0$: $$y = 2 \times 0 \times (0 + 3) \times (0 - 2) \times (0 - 4) = 0$$ So the y-intercept is 0. **End behavior:** Leading term is $2x^4$ (since multiplying all $x$ terms gives $x^4$ and coefficient 2). - Since degree is even and leading coefficient positive, as $x \to \pm \infty$, $y \to +\infty$. 3. **Analyze the second polynomial:** - The function is $y = (x^2 + 3)(x^2 - 4)$. **Degree:** Each factor is degree 2, so total degree is $2 + 2 = 4$. **Zeros and types:** Set each factor equal to zero: - $x^2 + 3 = 0 \Rightarrow x^2 = -3$ no real zeros here. - $x^2 - 4 = 0 \Rightarrow x^2 = 4 \Rightarrow x = \pm 2$ So zeros are $x = -2$ and $x = 2$, each with multiplicity 1 (since factors are linear in $x^2$). **Y-intercept:** Evaluate $y$ at $x=0$: $$y = (0^2 + 3)(0^2 - 4) = 3 \times (-4) = -12$$ So the y-intercept is $-12$. **End behavior:** Leading term is $x^2 \times x^2 = x^4$ with coefficient 1 (positive). - Since degree is even and leading coefficient positive, as $x \to \pm \infty$, $y \to +\infty$. 4. **Summary:** | Function | Degree | Zeros | Y-intercept | End Behavior | |---|---|---|---|---| | $2x(x+3)(x-2)(x-4)$ | 4 | $x=0,-3,2,4$ (all simple) | 0 | $y \to +\infty$ as $x \to \pm \infty$ | | $(x^2+3)(x^2-4)$ | 4 | $x=\pm 2$ (simple) | -12 | $y \to +\infty$ as $x \to \pm \infty$ | 5. **Graph shapes and math description:** - Both are quartic polynomials with positive leading coefficients. - The first has four distinct real roots, so the graph crosses the x-axis four times. - The second has two real roots and two complex roots, so it crosses the x-axis twice. - Both graphs rise to positive infinity on both ends. Final answers: - For $y = 2x(x + 3)(x - 2)(x - 4)$: - Degree: 4 - Zeros: $-3, 0, 2, 4$ (all simple) - Y-intercept: 0 - End behavior: $y \to +\infty$ as $x \to \pm \infty$ - For $y = (x^2 + 3)(x^2 - 4)$: - Degree: 4 - Zeros: $-2, 2$ (simple) - Y-intercept: -12 - End behavior: $y \to +\infty$ as $x \to \pm \infty$