These are the questions for task 1: (i) Identify the turning points, zeros, and x-intercepts.

(ii) Do you find any point or zero that has a multiplicity in the graph? If so, specify them with multiplicity and explain the reason.

(iii) Identify the degree and the polynomial, as well as the domain in which the polynomial is increasing and decreasing.

(iv) Do we have a local maximum/minimum here? If yes, find them.

(v) Find the remainder when the polynomial is divided by x-4.

Let's break down the polynomial for Task 1, \(2x + 3y = 6\), into the following components for the questions provided:

### Task 1: Polynomial Analysis

Given polynomial: \(2x + 3y = 6\)

First, we express \(y\) in terms of \(x\):
\[ 3y = 6 - 2x \]
\[ y = \frac{6 - 2x}{3} \]
\[ y = 2 - \frac{2x}{3} \]

So the function we are dealing with is:
\[ y = -\frac{2}{3}x + 2 \]

This is a linear equation. However, since the questions seem to pertain more to a polynomial (usually nonlinear), I'll make some assumptions about the type of polynomial expected and provide answers accordingly.

### (i) Identify the turning points, zeros, and x-intercepts.

#### Turning Points:
For linear equations, there are no turning points. Turning points exist in polynomials of degree 2 or higher.

#### Zeros and X-intercepts:
To find the x-intercepts, we set \(y = 0\):
\[ 0 = -\frac{2}{3}x + 2 \]
\[ \frac{2}{3}x = 2 \]
\[ x = 3 \]

So, the zero and x-intercept are:
\[ x = 3 \]

### (ii) Do you find any point or zero that has a multiplicity in the graph? If so, specify them with multiplicity and explain the reason.

For linear equations, zeros do not have multiplicity greater than 1. Multiplicity typically pertains to higher-degree polynomials where a root might be repeated.

### (iii) Identify the degree and the polynomial, as well as the domain in which the polynomial is increasing and decreasing.

The polynomial \(y = -\frac{2}{3}x + 2\) is of degree 1.

#### Increasing/Decreasing:
- **Increasing:** None, as the slope \(-\frac{2}{3}\) is negative.
- **Decreasing:** The polynomial is decreasing for all \(x\).

### (iv) Do we have a local maximum/minimum here? If yes, find them.

For linear equations, there are no local maxima or minima. These features exist in polynomials of degree 2 or higher.

### (v) Find the remainder when the polynomial is divided by \(x-4\).

Since the polynomial is linear, \(y = -\frac{2}{3}x + 2\), the remainder when divided by any binomial of the form \(x - a\) is simply the value of the polynomial at \(x = a\).

To find the remainder when the polynomial is divided by \(x - 4\), we evaluate the polynomial at \(x = 4\):
\[ y = -\frac{2}{3}(4) + 2 \]
\[ y = -\frac{8}{3} + 2 \]
\[ y = -\frac{8}{3} + \frac{6}{3} \]
\[ y = -\frac{2}{3} \]

Thus, the remainder is:
\[ -\frac{2}{3} \]

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Would you like further details on any specific point, or do you have any other questions? Here are some additional questions you might be interested in asking:

1. How can I graph a linear function?
2. What are the key characteristics of polynomial degrees and their graphs?
3. How can I identify multiplicity in higher-degree polynomials?
4. What are the differences between local and absolute maxima/minima?
5. How do polynomial division and the Remainder Theorem work?

**Tip:** When dealing with polynomials, understanding the graphical behavior (like turning points and end behavior) based on the degree and leading coefficients can help predict and analyze their properties effectively.

Explore the analysis of polynomials, focusing on identifying turning points, zeros, polynomial degrees, and their behavior. Learn how to apply the Remainder Theorem to find the remainder when a polynomial is divided by x-4. Suitable for high school students studying algebra.

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