**Part I — Identifying Key Features of Polynomial Functions** 1. Given each polynomial function, find its factored form, leading coefficient, degree, and end behavior. 2. For $(x - 2)(x^2 + 3x + 4)$: - Factored form: Already factored. - Leading coefficient: The $x^2$ term in second factor has leading coefficient 1, multiplied by 1 from $(x - 2)$, so overall positive. - Degree: $1 + 2 = 3$ (since multiplied linear by quadratic). - End behavior: - Leading term approx $x^3$ positive, so as $x \to -\infty$, $f(x) \to -\infty$ (falls left). - As $x \to \infty$, $f(x) \to \infty$ (rises right). 3. $(3x + 1)(x - 4)(x + 2)$: - Factored form: given. - Leading coefficient: $3 \times 1 \times 1 = 3 > 0$ positive. - Degree: $1 + 1 + 1 = 3$ cubic. - End behavior: cubic with positive leading coefficient. - Left: $f(x) \to -\infty$. - Right: $f(x) \to \infty$. 4. $-x^3 + 5x^2 - 3x + 2$: - Factored form: not given; leading term is $-x^3$ so leading coefficient $-1$. - Degree: 3. - End behavior: - Left: since degree odd and leading coefficient negative, $f(x) \to \infty$. - Right: $f(x) \to -\infty$. 5. $4x^4 - 2x^2 + x - 6$: - Leading coefficient 4, positive. - Degree 4 (even). - End behavior even degree positive leading: - Left: $\infty$ - Right: $\infty$ 6. $(x + 5)(x - 3)(x - 1)$: - Leading coefficient: $1\times 1\times 1 = 1 > 0$. - Degree 3. - End behavior odd degree positive leading: - Left: $-\infty$ - Right: $\infty$ 7. $2(x - 1)^2(x + 4)$: - Degree: $(2) + 1 = 3$ (quadratic factor squared plus linear). - Leading coefficient: $2\times 1^2 \times 1 = 2 > 0$. - End behavior odd degree positive: - Left: $-\infty$ - Right: $\infty$ 8. $-3x^5 + 7x^3 - x + 10$: - Degree 5. - Leading coefficient $-3 < 0$. - End behavior odd degree negative leading: - Left: $\infty$ - Right: $-\infty$ 9. $(x - 2)^3(x + 1)^2$: - Degree $3 + 2 = 5$. - Leading coefficient: $1^3 \times 1^2 = 1 > 0$. - End behavior odd degree positive: - Left: $-\infty$ - Right: $\infty$ 10. $5x^4 - 2x^3 + 6x - 1$: - Degree 4, leading coefficient 5 positive. - End behavior even degree positive: - Left: $\infty$ - Right: $\infty$ 11. $-2x^6 + 4x^3 - x^2 + 9$: - Degree 6 even. - Leading coefficient $-2 < 0$ negative. - End behavior even degree negative: - Left: $-\infty$ - Right: $-\infty$ 12. $(x + 2)^4(x - 3)$: - Degree $4 + 1 = 5$ odd. - Leading coefficient $1^4 \times 1 = 1 > 0$ positive. - End behavior odd degree positive: - Left: $-\infty$ - Right: $\infty$ 13. $(x - 1)^2(x + 1)^2(x - 2)$: - Degree $2 + 2 + 1 = 5$ odd. - Leading coefficient $1^2 \times 1^2 \times 1 = 1 > 0$ positive. - End behavior odd positive: - Left: $-\infty$ - Right: $\infty$ **Part II — Zeros, Multiplicity, Behavior** A. For each function: 1. $(x - 2)^2(x + 1)$: - Zeros: 2, -1 - Multiplicity: 2 (at 2), 1 (at -1) - Characteristic: Even multiplicity at 2 (touches x-axis), odd multiplicity at -1 (crosses) - Behavior: Graph touches x-axis at 2 and bounces, crosses at -1. 2. $(x + 3)(x - 1)^3$: - Zeros: -3, 1 - Multiplicity: 1 (at -3), 3 (at 1) - Characteristic: Crosses at -3 linear, crosses with flattening at 1 cubic. 3. $(x - 4)(x + 2)^2(x - 1)$: - Zeros: 4, -2, 1 - Multiplicity: 1 (4), 2 (-2), 1 (1) - Behavior: crosses at 4 and 1, touches and bounces at -2. 4. $(x + 1)^4(x - 2)$: - Zeros: -1, 2 - Multiplicity: 4 (at -1), 1 (at 2) - Behavior: touches and flattens at -1, crosses at 2. 5. $x(x - 3)^2(x + 4)$: - Zeros: 0, 3, -4 - Multiplicity: 1 (0), 2 (3), 1 (-4) - Behavior: crosses at 0 and -4, touches and bounces at 3. 6. $(x + 2)^3(x - 1)(x - 5)$: - Zeros: -2, 1, 5 - Multiplicity: 3 (-2), 1 (1), 1 (5) - Behavior: crosses with flattening at -2, crosses at 1 and 5. B. Compute $P(x)$ values: 1. $P(x) = x(x - 2)(x + 1)^2(x - 3)$: - For each $x$: - $P(-3) = (-3)(-5)(-2)^2(-6) = (-3)(-5)(4)(-6) = (-3 imes -5) imes 4 imes -6 = 15 imes 4 imes -6 = 60 imes -6 = -360$ - $P(-1) = (-1)(-3)(0)^2(-4) = 0$ - $P(0) = 0$ - $P(1) = (1)(-1)(2)^2(-2) = 1 imes (-1) imes 4 imes (-2) = (-1) imes 4 imes (-2) = -4 imes (-2) = 8$ - $P(2) = (2)(0)(3)^2(-1) = 0$ - $P(3) = (3)(1)(4)^2(0) = 0$ 2. $P(x) = (x + 2)(x - 1)(x - 3)^2$: - $P(-2) = 0$ - $P(0) = (2)(-1)(-3)^2 = 2 imes -1 imes 9 = -18$ - $P(1) = (3)(0)(-2)^2 = 0$ - $P(2) = (4)(1)(-1)^2 = 4 imes 1 imes 1 = 4$ - $P(3) = (5)(2)(0)^2 = 0$ - $P(4) = (6)(3)(1)^2 = 6 imes 3 imes 1 = 18$ 3. $P(x) = (x - 4)^2(x + 1)^3$ (values not requested to compute).