where:

• $n$ is a non-negative integer (the degree of the polynomial)
• a_n, a_{n-1, ..., a_0 are real numbers (the coefficients)
• $a_n \neq 0$ (the leading coefficient)

Degree and Leading Coefficient

The degree is the highest power of $x$ in the polynomial.

The leading coefficient is the coefficient of the term with the highest power.

Example: $f(x) = 3x^4 - 2x^3 + 5x - 7$

• Degree: 4
• Leading coefficient: 3

End Behavior

End behavior describes what happens to $f(x)$ as $x \to \infty$ and $x \to -\infty$.

End Behavior Rules

The end behavior depends on:

1. The degree (even or odd)
2. The sign of the leading coefficient (positive or negative)

Key Concepts

1. Even degree, positive leading coefficient: Both ends go up (U-shape)
2. Even degree, negative leading coefficient: Both ends go down (upside-down U)
3. Odd degree, positive leading coefficient: Left goes down, right goes up (/)
4. Odd degree, negative leading coefficient: Left goes up, right goes down ()

Zeros and Multiplicity

The zeros (or roots) of a polynomial are the values of $x$ where $f(x) = 0$.

The multiplicity of a zero is how many times that factor appears.

• Odd multiplicity: The graph crosses the x-axis
• Even multiplicity: The graph touches the x-axis and turns around

Practice Strategy

To analyze a polynomial:

1. Identify the degree and leading coefficient
2. Determine the end behavior
3. Find the zeros and their multiplicities
4. Sketch the general shape of the graph

Part (c): For end behavior, we look at the degree and leading coefficient:

• Degree: 5 (odd)
• Leading coefficient: -2 (negative)

For a polynomial with odd degree and negative leading coefficient:

• As $x \to \infty$, $f(x) \to -\infty$
• As $x \to -\infty$, $f(x) \to \infty$

In words: The graph rises to the left and falls to the right.

Step 2: Determine the multiplicity of each zero.

• $x = 0$: multiplicity 2 (even)
• $x = 3$: multiplicity 3 (odd)
• $x = -1$: multiplicity 1 (odd)

Step 3: Describe behavior at each zero.

• At $x = 0$: The graph touches the x-axis and turns around (even multiplicity)
• At $x = 3$: The graph crosses the x-axis (odd multiplicity)
• At $x = -1$: The graph crosses the x-axis (odd multiplicity)

Step 4: Determine overall end behavior.

• Degree: $2 + 3 + 1 = 6$ (even)
• Leading coefficient: positive (from expanding, the leading term is $x^6$)
• End behavior: Both ends go to $\infty$

Answer: Zeros at $x = -1$ (crosses), $x = 0$ (touches), and $x = 3$ (crosses)

Since the degree is 4 and we only have 3 zeros (each multiplicity 1), we need one more factor. Let's call it $(x - r)$ where $r$ is unknown.

General form: $g(x) = a(x + 2)(x - 1)(x - 3)(x - r)$

where $a$ is a positive constant.

Part (b): We use the point $(0, -12)$ to find $a$ and $r$.

However, with two unknowns and one equation, we need more information. Let's assume the fourth zero is actually a repeated zero. If $x = -2$ has multiplicity 2:

$g(x) = a(x + 2)^2(x - 1)(x - 3)$

Substituting $(0, -12)$: $-12 = a(2)^2(-1)(-3)$ $-12 = a(4)(3)$ $-12 = 12a$ $a = -1$

But we need $a$ positive! So let's try $x = 1$ with multiplicity 2:

$g(x) = a(x + 2)(x - 1)^2(x - 3)$

$-12 = a(2)(1)^2(-3)$ $-12 = a(2)(-3) = -6a$ $a = 2$

Therefore: $g(x) = 2(x + 2)(x - 1)^2(x - 3)$

Part (c): Degree 4 (even), positive leading coefficient:

• As $x \to \infty$, $g(x) \to \infty$
• As $x \to -\infty$, $g(x) \to \infty$

The graph rises on both ends.

Step 2: Determine the sign of the leading coefficient.

Total degree: $2 + 1 + 1 = 4$ (even)

For both ends to go to $-\infty$ with an even degree, we need a negative leading coefficient.

So $a < 0$. We can choose $a = -1$.

$f(x) = -(x + 2)^2(x - 1)(x - 4)$

Step 3: Verify (optional - expand to check).

If we expand: $f(x) = -(x + 2)^2(x^2 - 5x + 4)$

The leading term will be $-x^4$, confirming our negative leading coefficient.

Answer: $f(x) = -(x + 2)^2(x - 1)(x - 4)$ or any negative constant multiple