Question

polynomial operations maze
directions: start in the box labeled \start here.\ solve the problem in the box, color the correct
solution and then solve the box adjacent to the solution. continue until you reach \finished.\
start here!
$(3x^4 - 8x) + (2x - 4x^4)$ $-x^4 - 6x$ $(2x - 1)(7x - 2)$ $14x^2 - 11x + 2$ $x^2 - (x^2 + 9x - 3)$
$x^4 + 6x$ $5x^4 - 12x$ $9x^2 - 3$ $2x^2 - 9x - 3$ $-9x + 3$
$dfrac{3x}{3x^2}$ $x^4 - 6x^3 - 3x$ $(6x - 5x^4) - (4x^4 + 6x^3 + 3x)$ $12x^2 - 6x$ $3x(4x - 2)$
$1$ $-9x^4 - 6x^3 + 3x$ $-x^4 + 9x$ $-2x + x^3 - 3x$ $7x^2 - 5x$
$(3x - 7)^2$ $9x^2 - 42x + 49$ $dfrac{7x^3}{14x}$ $2x^{-1}$ $(5x^3 - 5x - 1) + (5x + 6)$
$9x^2 - 49$ $3x^2 - 14$ $dfrac{1}{2}x^2$ $8x^3 + 55x^2 - 8x - 7$ $5x^3 + 5$
$(5x^2 + 1) + (2 - x)$ $7x^2 - x + 3$ $(x + 7)(8x^2 - x - 1)$ $8x^2 + 6$ $2x(5x^2 + x)$
$5x^2 + 2$ $x^2 - 8x + 12$ $-7x^2 + x$ $9x^3 + 7x - 7$ $10x^3 + 2x^2$
$(x - 4)(x + 4)$ $x^2 + 16$ finished! $4x^3 + 3x + 7$ $(7x^2 - x + 4) - (3x^2 - 4x - 3)$

Explanation:

Step 1: Start with the first problem

We start with the "Start here" box: \((3x^{4}-8x)+(2x - 4x^{4})\)
Combine like terms: \(3x^{4}-4x^{4}-8x + 2x=-x^{4}-6x\)

Step 2: Move to the adjacent box with \(-x^{4}-6x\)

Now, look at the adjacent boxes. The box with \(-x^{4}-6x\) is adjacent. Wait, no, we need to find the box adjacent to our solution \(-x^{4}-6x\). Wait, actually, the first solution is \(-x^{4}-6x\), now we look for the box adjacent to this solution. Wait, maybe I made a mistake. Wait, the first problem is \((3x^{4}-8x)+(2x - 4x^{4})\), let's re - calculate:

\((3x^{4}-8x)+(2x - 4x^{4})=3x^{4}-4x^{4}-8x + 2x=-x^{4}-6x\)

Now, the box with \(-x^{4}-6x\) is a path. Now, from \(-x^{4}-6x\), we look at adjacent boxes. Wait, maybe the next step is to solve the next problem. Wait, no, the maze is solved by solving the problem in the box, coloring the correct solution (the result of the problem) and then moving to the adjacent box with that solution.

Wait, let's correct. The first box is "Start here" with \((3x^{4}-8x)+(2x - 4x^{4})\). We solve it:

\(3x^{4}-4x^{4}-8x + 2x=-x^{4}-6x\)

Now, we look for the box (adjacent to the "Start here" box) that has \(-x^{4}-6x\) as its content. Then we move to that box. Then we solve the problem in that box? Wait, no, the directions say: "Solve the problem in the box, color the correct solution and then solve the box adjacent to the solution". Wait, maybe it's: solve the problem in the current box, get the solution, then find the adjacent box that has that solution, and then solve the problem in that adjacent box.

Let's start over:

1. **First box (Start here):** \((3x^{4}-8x)+(2x - 4x^{4})\)
- Combine like terms: \(3x^{4}-4x^{4}-8x + 2x=-x^{4}-6x\)
- Now, find the adjacent box with \(-x^{4}-6x\). The box to the right of "Start here" has \(-x^{4}-6x\). So we move to that box.

2. **Second box (with \(-x^{4}-6x\))? Wait, no, the box to the right of "Start here" is \(-x^{4}-6x\), but that's a solution, not a problem. Wait, maybe the first problem is \((3x^{4}-8x)+(2x - 4x^{4})\), solution is \(-x^{4}-6x\). Then we look for the box adjacent to the "Start here" box that has \(-x^{4}-6x\) as its value. Then we move to that box, and then solve the problem in that box? Wait, no, the next box to solve is the one adjacent to the solution. Wait, maybe the maze is structured such that each box has a problem, and the solution of the problem is the value of an adjacent box. So we solve the problem in the current box, then move to the adjacent box with that solution, and repeat.

Let's try again:

- **Box 1 (Start here):** \((3x^{4}-8x)+(2x - 4x^{4})\)
- Solution: \(3x^{4}-4x^{4}-8x + 2x=-x^{4}-6x\)
- Adjacent boxes to "Start here" are: right (\(-x^{4}-6x\)), down (\(x^{4}+6x\)), and a diagonal (\(5x^{4}-12x\)). The solution \(-x^{4}-6x\) is in the right - adjacent box. So we move to the right - adjacent box (which has \(-x^{4}-6x\) as its content, but actually, that box is a "solution" box, and the next problem is in the box adjacent to this solution box. Wait, I think I misinterpreted. Let's look at the second row. The box in the middle of the second row is \((6x - 5x^{4})-(4x^{4}+6x^{3}+3x)\). Wait, no, let's take a simple problem. Let's take the "Start here" problem:

\((3x^{4}-8x)+(2x - 4x^{4})=3x^{4}-4x^{4}-8x + 2x=-x^{4}-6x\)

Now, the box to the right of "Start here" is \(-x^{4}-6x\) (a solution), and the box to the right of that is \((2x - 1)(7x - 2)\). Wait, no, the directions say "solve the problem in the box, color the correct solution and then solve the box adjacent to the solution". So first, solve the problem in the current box (get the solution), then find the adjacent box that has that solution (color it), and then solve the problem in that adjacent box.

Let's take another problem, say the "Start here" problem:

Problem: \((3x^{4}-8x)+(2x - 4x^{4})\)
Solution: \(-x^{4}-6x\)
Now, find the box adjacent to the "Start here" box that has \(-x^{4}-6x\) as its value. The box to the right of "Start here" has \(-x^{4}-6x\). So we color that box (the one with \(-x^{4}-6x\)) and then solve the problem in the box adjacent to \(-x^{4}-6x\). The box adjacent to \(-x^{4}-6x\) (to the right) is \((2x - 1)(7x - 2)\).

Now, solve \((2x - 1)(7x - 2)\):

- Use the distributive property (FOIL method): \(2x\times7x-2x\times2-1\times7x + 1\times2\)
- \(14x^{2}-4x-7x + 2=14x^{2}-11x + 2\)

Now, find the box adjacent to \((2x - 1)(7x - 2)\) that has \(14x^{2}-11x + 2\) as its value. The box to the right of \((2x - 1)(7x - 2)\) has \(14x^{2}-11x + 2\). So we color that box and move to the box adjacent to it.

The box adjacent to \(14x^{2}-11x + 2\) (to the right) is \(x^{2}-(x^{2}+9x - 3)\).

Solve \(x^{2}-(x^{2}+9x - 3)\):

- Distribute the negative sign: \(x^{2}-x^{2}-9x + 3=-9x + 3\)

Find the box adjacent to \(x^{2}-(x^{2}+9x - 3)\) that has \(-9x + 3\) as its value. The box below \(x^{2}-(x^{2}+9x - 3)\) has \(-9x + 3\). Color that box and move to the box adjacent to it.

The box adjacent to \(-9x + 3\) (below) is \(3x(4x - 2)\).

Solve \(3x(4x - 2)\):

- Distribute: \(3x\times4x-3x\times2 = 12x^{2}-6x\)

Find the box adjacent to \(3x(4x - 2)\) that has \(12x^{2}-6x\) as its value. The box to the left of \(3x(4x - 2)\) has \(12x^{2}-6x\). Color that box and move to the box adjacent to it.

The box adjacent to \(12x^{2}-6x\) (to the left) is \((6x - 5x^{4})-(4x^{4}+6x^{3}+3x)\).

Solve \((6x - 5x^{4})-(4x^{4}+6x^{3}+3x)\):

- Distribute the negative sign: \(6x - 5x^{4}-4x^{4}-6x^{3}-3x\)
- Combine like terms: \(-9x^{4}-6x^{3}+3x\)

Find the box adjacent to \((6x - 5x^{4})-(4x^{4}+6x^{3}+3x)\) that has \(-9x^{4}-6x^{3}+3x\) as its value. The box below \((6x - 5x^{4})-(4x^{4}+6x^{3}+3x)\) has \(-9x^{4}-6x^{3}+3x\). Color that box and move to the box adjacent to it.

The box adjacent to \(-9x^{4}-6x^{3}+3x\) (below) is \((5x^{3}-5x - 1)+(5x + 6)\).

Solve \((5x^{3}-5x - 1)+(5x + 6)\):

- Combine like terms: \(5x^{3}-5x + 5x-1 + 6=5x^{3}+5\)

Find the box adjacent to \((5x^{3}-5x - 1)+(5x + 6)\) that has \(5x^{3}+5\) as its value. The box below \((5x^{3}-5x - 1)+(5x + 6)\) has \(5x^{3}+5\). Color that box and move to the box adjacent to it.

The box adjacent to \(5x^{3}+5\) (below) is \(2x(5x^{2}+x)\).

Solve \(2x(5x^{2}+x)\):

- Distribute: \(2x\times5x^{2}+2x\times x=10x^{3}+2x^{2}\)

Find the box adjacent to \(2x(5x^{2}+x)\) that has \(10x^{3}+2x^{2}\) as its value. The box below \(2x(5x^{2}+x)\) has \(10x^{3}+2x^{2}\). Color that box and move to the box adjacent to it.

The box adjacent to \(10x^{3}+2x^{2}\) (below) is \((7x^{2}-x + 4)-(3x^{2}-4x - 3)\).

Solve \((7x^{2}-x + 4)-(3x^{2}-4x - 3)\):

- Distribute the negative sign: \(7x^{2}-x + 4-3x^{2}+4x + 3\)
- Combine like terms: \(4x^{2}+3x + 7\)

Find the box adjacent to \((7x^{2}-x + 4)-(3x^{2}-4x - 3)\) that has \(4x^{2}+3x + 7\) as its value. The box to the left of \((7x^{2}-x + 4)-(3x^{2}-4x - 3)\) has \(4x^{2}+3x + 7\). Color that box and move to the box adjacent to it.

The box adjacent to \(4x^{2}+3x + 7\) (to the left) is "Finished!".

Answer:

The path goes through the following problem - solution steps:

1. \((3x^{4}-8x)+(2x - 4x^{4})=-x^{4}-6x\)
2. \((2x - 1)(7x - 2)=14x^{2}-11x + 2\)
3. \(x^{2}-(x^{2}+9x - 3)=-9x + 3\)
4. \(3x(4x - 2)=12x^{2}-6x\)
5. \((6x - 5x^{4})-(4x^{4}+6x^{3}+3x)=-9x^{4}-6x^{3}+3x\)
6. \((5x^{3}-5x - 1)+(5x + 6)=5x^{3}+5\)
7. \(2x(5x^{2}+x)=10x^{3}+2x^{2}\)
8. \((7x^{2}-x + 4)-(3x^{2}-4x - 3)=4x^{2}+3x + 7\)

And finally reaches "Finished!".