# 4.1

Examples of polynomials:

$$2x^3 - 5x^2 + 4x + 1$$
$$7x^4 + 3x^3 - x^2 - x + 5$$
$$2x^9 - 4x^4 - 2x^2 - 1$$
$$7x + 2$$
$$8x^2 - 6x - 1$$

- Sums/differences of multiples of powers of $x$.
- The powers on $x$ are $0, 1, 2, 3, 4, 5, \dots$
- No $x^{1.5}$ or $x^{-3}$, etc.
- Recall $x^1 = x$ and $x^0 = 1$.
- Linear and quadratic are special types of polynomials.

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Terms can be ordered as we like:

$$2x^3 - 5x^2 + 4x + 1$$
$$4x + 1 - 5x^2 + 2x^3$$
$$-5x^2 + 1 + 2x^3 + 4x$$
$$1 + 4x - 5x^2 + 2x^3$$

are all the same polynomial.

- **Degree of polynomial** is the highest power of $x$ appearing.
- **Leading term** is the term of that degree.
- **Leading coefficient** is the coefficient of the leading term.

### Examples. Give degree, leading term, and leading coefficient for
---
$$2x^3-5x^2+4x+1$$

Degree: $3$

Leading Term: $2x^3$

Leading Coefficient: $2$

---

$$2x^2-4x^4-2x^9-1$$

Degree: $9$

Leading Term: $-2x^9$

Leading Coefficient: $-2$

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## End behavior.

- For any polynomial, as $x$ gets very large positive or large negative, $y$
also gets very large positive or negative, all depending on the {\it leading term}
because its values are much larger than the other terms for large positive or negative
$x$.

- The graphs must then roughly look like one of these:


- If the leading term is $ax^n$, then we may figure which it is by:

$a>0$, $n$ even

$a<0$, $n$ even

$a>0$, $n$ odd

$a<0$, $n$ odd

- Summary Table Regarding End Behavior based off the Leading Term Test:

| | Even Degree | Odd Degree |
|---|---|---|
| Positive Leading Coefficient |  rises on both sides |  rises to the right, falls to the left|
| Negative Leading Coefficient |  falls on both sides |  rises to the left, falls to the right|

- We can see this by plugging in large values of $x$ like $\pm 10$
and seeing whether $ax^n$ is positive or negative. If $ax^n>0$ then the graph
goes up. If $ax^n<0$ then the graph goes down.

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### Example. Determine the end behavior for $$2x^2-4x^4-2x^9-1$$

Degree: $9$ (odd)

Leading Term: $-2x^9$

Leading Coefficient: $-2$ (negative)


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### Example. Determine the end behavior for $$7x^4+3x^3-x^2-x+5$$

Degree: $4$ (even)

Leading Term: $7x^4$

Leading Coefficient: $7$ (positive)


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### Example. Determine the end behavior for $$-3x^4+7x^3-2x^2-x+5$$

Degree: $4$ (even)

Leading Term: $-3x^4$

Leading Coefficient: $-3$ (negative)


---

### Example. Determine the end behavior for $$7x^5-2x^3-x+4+3x^7$$

Degree: $7$ (odd)

Leading Term: $3x^7$

Leading Coefficient: $3$ (positive)


---

### **What value of $x$ makes a polynomial zero?**

- This gives $x$ intercepts for graphs.

- We can do this by factoring, which is often difficult.

- Here we will only look at polynomials which have already been factored.

- A factor $(x-a)$ produces a value $x=a$, just as with quadratics.

- The factor might be raised to a power and look like $(x-a)^n$. The zero
is $x=a$ and we say it has **multiplicity** $n$.

### Example. Find the zeroes and multiplicities for the function $$f(x) = (x+4)^3 (x-2) (x-3)^2$$

| Factor | Zero | Multiplicity |
|----------------|---------------|------------------|
| $(x+4)^3$ | $x = -4$ | 3 |
| $(x-2)$ | $x = 2$ | 1 |
| $(x-3)^2$ | $x = 3$ | 2 |

**Conclusion:**
- The zero at $x = -4$ has a multiplicity of 3.
- The zero at $x = 2$ has a multiplicity of 1.
- The zero at $x = 3$ has a multiplicity of 2.

### Example. Find zeroes and multiplicities for $$f(x)=(x+7)^2(x-3)^4(x-5)$$

| Factor | Zero | Multiplicity |
|----------------|---------------|------------------|
| $(x+7)^2$ | $x = -7$ | 2 |
| $(x-3)^4$ | $x = 3$ | 4 |
| $(x-5)$ | $x = 5$ | 1 |

**Conclusion:**
- The zero at $x = -7$ has a multiplicity of 2.
- The zero at $x = 3$ has a multiplicity of 4.
- The zero at $x = 5$ has a multiplicity of 1.

### Example. Find zeroes and multiplicities for $$f(x)=(x+a)^n(x-b)^m(x-c)^p$$

| Factor | Zero | Multiplicity |
|----------------|---------------|------------------|
| $(x+a)^n$ | $x = -a$ | n |
| $(x-b)^m$ | $x = b$ | m |
| $(x-c)^p$ | $x = c$ | p |

**Conclusion:**
- The zero at $x = -a$ has a multiplicity of n.
- The zero at $x = b$ has a multiplicity of m.
- The zero at $x = c$ has a multiplicity of p.