1.Say I am dividing 4xcubed + x + 7 by x+15 do i need to add in an x squared or can I divide it how it is? Sorry that is confusing

2. I have a question that says, "on what bounded intervals is the function increasing/decreasing" (on a graph). I'm not sure what it is asking for help please!

1. To divide a polynomial by another polynomial, such as dividing 4x^3 + x + 7 by x + 15, you can use long division or synthetic division. In this case, let's use long division:

Start by dividing the first term of the dividend (4x^3) by the first term of the divisor (x). This gives you 4x^2. Place this quotient above the line.

Next, multiply the entire divisor (x + 15) by the quotient you just found (4x^2). This gives you 4x^3 + 60x^2. Subtract this result from the original dividend (4x^3 + x + 7).

You should be left with -59x^2 + x + 7 in the dividend. Now, repeat the process by dividing the first term of the new dividend (-59x^2) by the first term of the divisor (x). This gives you -59x. Place this quotient above the line.

Multiply the entire divisor (x + 15) by the new quotient (-59x). This gives you -59x^2 - 885x. Subtract this result from the new dividend (-59x^2 + x + 7).

Now you should have 886x + 7 as the new dividend. Divide the first term (886x) by the first term (x) of the divisor to get 886. Place this quotient above the line.

Multiply the entire divisor (x + 15) by the quotient (886). This gives you 886x + 13380. Subtract this result from the new dividend (886x + 7).

The result is -13373 in the dividend, which cannot be further divided by the divisor. So, the final result of dividing 4x^3 + x + 7 by x + 15 is the quotient 4x^2 - 59x + 886 and the remainder -13373.

2. When a question asks about the bounded intervals where a function is increasing/decreasing, it is essentially asking for the intervals on the x-axis where the function is either going up or going down.

To determine whether a function is increasing or decreasing on a graph, you need to look at the slope of the function at different points. If the slope is positive, the function is increasing. If the slope is negative, the function is decreasing.

To identify the bounded intervals where the function is increasing/decreasing, you should follow these steps:
1. Locate the critical points of the function. These are the points where the function changes from increasing to decreasing or vice versa. Critical points occur where the slope is zero or undefined.
2. Evaluate the slope of the function between each pair of critical points to determine whether the function is increasing or decreasing in those intervals.
3. Use inequality notation to describe the intervals on which the function is increasing (positive slope) or decreasing (negative slope). For example, if the function is increasing from x = 1 to x = 5, you would write the interval as (1, 5).

By analyzing the slope and critical points, you can determine the bounded intervals where the function is increasing/decreasing on a graph.