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.