QuestionSimplify. $\left(-w^{2}+7 w-4\right)-\left(5 w^{2}+3 w+2\right)$

Studdy Solution

STEP 1

What is this asking? We're taking one polynomial and subtracting another polynomial from it, which basically means combining like terms. Watch out! Don't forget to distribute the negative sign when subtracting the second polynomial!
This is where a lot of people trip up.

STEP 2

1. Distribute the negative sign
2. Combine like terms

STEP 3

Alright, let's kick things off by distributing that negative sign to every term in the second set of parentheses.
Think of it like multiplying each term by $-1$ .
This is super important because subtracting is the same as adding the opposite.

STEP 4

So, we have $\left(-w^{2}+7 w-4\right) + (-1) \cdot \left(5 w^{2}+3 w+2\right).$

STEP 5

Distributing the $-1$ gives us $\left(-w^{2}+7 w-4\right)+\left(-5 w^{2}-3 w-2\right).$ See how all the signs flipped in the second polynomial?
That's the magic of distributing the negative!

STEP 6

Now for the fun part!
We're going to combine like terms.
Remember, "like terms" are terms that have the same variable raised to the same power.
It's like sorting your socks – you want to put all the matching ones together.

STEP 7

Let's start with the * $\text{w}^2$ * terms.
We have $-w^{2}$ and $-5w^{2}$ .
Combining these gives us $(-1-5)w^2 = -6w^{2}$ .

STEP 8

Next up, the * $\text{w}$ * terms.
We have $7w$ and $-3w$ .
Combining these gives us $(7-3)w = 4w$ .

STEP 9

Finally, let's tackle the constant terms.
We have $-4$ and $-2$ .
Combining these gives us $(-4)+(-2) = -6$ .

STEP 10

Putting it all together, our simplified expression is $-6w^{2}+4 w-6$ .
Boom!

STEP 11

$-6w^{2}+4 w-6$

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