Question

Write each polynomial function in standard form. For each function, find the degree, number of terms, and leading coefficient

(a) f(x)= - 3x²+2x, 5 exponet + 8x³-6+x⁴-3x²
(b) f(x) = 8x²+ 10x, 7 exponet - 7x³-x⁴
(c) f(x) = - x³+9x+12 - x⁴+ 5x²

STANDARD FORM. Degree. Number of Terms. Leading Coefficient

(a)

(b)

(c)




Use the Leading Coefficient a, and Degree n, of Polynomial Function to determine the End Behavior of the graph, Show with a sketched graph



(d) f(x) = - x 5 exponent + 2x⁴ + 3x³ + 2x² - 8x+9





(e) f(x) = 7x⁴- 4x³+ 7x²+10x-15




(f) f(x) = - x, 6 exponet + 7x, 5 exponent - x⁴+2x³+9x²-8x-2

Answer 1

Let's start by rewriting each polynomial in standard form, which means arranging the terms in descending order of their degree. After formatting the polynomial functions appropriately, we will determine the degree, number of terms, and leading coefficient for each.

### (a)
**Original:**
\[ f(x) = -3x^2 + 2x + 5 + 8x^3 - 6 + x^4 - 3x^2 \]

**Standard Form:**
Combining like terms:
\[ f(x) = x^4 + 8x^3 - 6 - 3x^2 - 3x^2 + 2x + 5 \]
\[ f(x) = x^4 + 8x^3 - 6 - 6x^2 + 2x + 5 \]
\[ f(x) = x^4 + 8x^3 - 6x^2 + 2x - 1 \]

**Degree:** 4
**Number of Terms:** 5
**Leading Coefficient:** 1

---

### (b)
**Original:**
\[ f(x) = 8x^2 + 10x - 7 - 7x^3 - x^4 \]

**Standard Form:**
Rearranging in descending order:
\[ f(x) = -x^4 - 7x^3 + 8x^2 + 10x - 7 \]

**Degree:** 4
**Number of Terms:** 5
**Leading Coefficient:** -1

---

### (c)
**Original:**
\[ f(x) = -x^3 + 9x + 12 - x^4 + 5x^2 \]

**Standard Form:**
Rearranging in descending order:
\[ f(x) = -x^4 - x^3 + 5x^2 + 9x + 12 \]

**Degree:** 4
**Number of Terms:** 5
**Leading Coefficient:** -1

---

### Summary of (a), (b), and (c)

| Polynomial | Standard Form | Degree | Number of Terms | Leading Coefficient |
|-------------|----------------------------------------|--------|-----------------|---------------------|
| (a) | \( x^4 + 8x^3 - 6x^2 + 2x - 1 \) | 4 | 5 | 1 |
| (b) | \( -x^4 - 7x^3 + 8x^2 + 10x - 7 \) | 4 | 5 | -1 |
| (c) | \( -x^4 - x^3 + 5x^2 + 9x + 12 \) | 4 | 5 | -1 |

---

### End Behavior

**For (d) f(x) = -x^5 + 2x^4 + 3x^3 + 2x^2 - 8x + 9:**

- **Degree:** 5 (odd)
- **Leading Coefficient:** -1 (negative)

**End Behavior:**
As \( x \to -\infty, f(x) \to +\infty \)
As \( x \to +\infty, f(x) \to -\infty \)

---

**For (e) f(x) = 7x^4 - 4x^3 + 7x^2 + 10x - 15:**

- **Degree:** 4 (even)
- **Leading Coefficient:** 7 (positive)

**End Behavior:**
As \( x \to -\infty, f(x) \to +\infty \)
As \( x \to +\infty, f(x) \to +\infty \)

---

**For (f) f(x) = -x^6 + 7x^5 - x^4 + 2x^3 + 9x^2 - 8x - 2:**

- **Degree:** 6 (even)
- **Leading Coefficient:** -1 (negative)

**End Behavior:**
As \( x \to -\infty, f(x) \to -\infty \)
As \( x \to +\infty, f(x) \to -\infty \)

---

### Summary for (d), (e), and (f)

| Polynomial | Degree | Leading Coefficient | End Behavior |
|-------------|--------|---------------------|--------------------------------------------|
| (d) | 5 | -1 | \( \left( \begin{array}{l} x \to -\infty \\ f(x) \to +\infty \end{array} \right) \), \( \left( \begin{array}{l} x \to +\infty \\ f(x) \to -\infty \end{array} \right) \) |
| (e) | 4 | 7 | \( \left( \begin{array}{l} x \to -\infty \\ f(x) \to +\infty \end{array} \right) \), \( \left( \begin{array}{l} x \to +\infty \\ f(x) \to +\infty \end{array} \right) \) |
| (f) | 6 | -1 | \( \left( \begin{array}{l} x \to -\infty \\ f(x) \to -\infty \end{array} \right) \), \( \left( \begin{array}{l} x \to +\infty \\ f(x) \to -\infty \end{array} \right) \) |

### Sketches
1. **For (d)**: A downward odd-degree polynomial starts high on the left and finishes low on the right.
2. **For (e)**: An upward even-degree polynomial starts up on both ends.
3. **For (f)**: A downward even-degree polynomial starts down on both ends.

You can sketch these based on the aforementioned behaviors.