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Complete the steps to find the sum. What is the sum?

1. Rewrite terms that are subtracted as the addition of the opposite.
[tex]\[ -7g^4 + 4g^3 - 3g^2 + 5g - 3 \][/tex]
[tex]\[ -4g^4 - 3g^3 + 4g^2 + 5g + 3 \][/tex]
[tex]\[ g^2 + (-4g^4) + 5g + 9 + (-3g^3) + 3g^2 + (-6) \][/tex]

2. Group like terms.
[tex]\[ -4g^4 + 4g^2 + 14g - 6 \][/tex]

3. Combine like terms.

4. Write the resulting polynomial in standard form.


Sagot :

Sure, let's break this down step-by-step:

## Step 1: Rewrite the terms that are subtracted as the addition of the opposite.
We are given the following polynomial expressions:
[tex]\[ -7g^4 + 4g^3 - 3g^2 + 5g - 3 \][/tex]
[tex]\[ -4g^4 - 3g^3 + 4g^2 + 5g + 3 \][/tex]
[tex]\[ g^2 + (-4g^4) + 5g + 9 + (-3g^3) + 3g^2 + (-6) \][/tex]
[tex]\[ -4g^4 + 4g^2 + 14g - 6 \][/tex]

Let’s rewrite any subtraction as addition of the opposite:
[tex]\[ -7g^4 + 4g^3 + (-3g^2) + 5g + (-3) \][/tex]
[tex]\[ -4g^4 + (-3g^3) + 4g^2 + 5g + 3 \][/tex]
[tex]\[ (-4g^4) + (-3g^3) + g^2 + 3g^2 + 5g + 9 + (-6) \][/tex]
[tex]\[ -4g^4 + 4g^2 + 14g + (-6) \][/tex]

## Step 2: Group like terms.
First, let's list out all terms grouped by their degree:
[tex]\[ \begin{align*} -7g^4 &+ 4g^3 + (-3g^2) + 5g + (-3) \\ &+ (-4g^4) + (-3g^3) + 4g^2 + 5g + 3 \\ &+ (-4g^4) + (-3g^3) + g^2 + 3g^2 + 5g + 9 + (-6) \\ &+ (-4g^4) + 4g^2 + 14g + (-6) \end{align} \][/tex]

Now, recombine them by sorting the terms according to their degree:
[tex]\[ \begin{align
} =& (-7g^4 + (-4g^4) + (-4g^4) + (-4g^4)) \\ &+ (4g^3 + (-3g^3) + (-3g^3) + (-3g^3)) \\ &+ ((-3g^2) + 4g^2 + g^2 + 3g^2 + 4g^2) \\ &+ \left( 5g + 5g + 5g + 14g \right) \\ &+ (-3 + 3 + 9 + (-6) + (-6)) \end{align*} \][/tex]

## Step 3: Combine like terms.
- For [tex]\( g^4 \)[/tex] terms:
[tex]\[ -7g^4 + (-4g^4) + (-4g^4) + (-4g^4) = -7g^4 - 12g^4 = -19g^4 \][/tex]

- For [tex]\( g^3 \)[/tex] terms:
[tex]\[ 4g^3 + (-3g^3) + (-3g^3) + (-3g^3) = 4g^3 - 9g^3 = -5g^3 \][/tex]

- For [tex]\( g^2 \)[/tex] terms:
[tex]\[ -3g^2 + 4g^2 + g^2 + 3g^2 + 4g^2 = 9g^2 - 3g^2 = 6g^2 \][/tex]

- For [tex]\( g \)[/tex] terms:
[tex]\[ 5g + 5g + 5g + 14g = 29g \][/tex]

- Constant terms:
[tex]\[ -3 + 3 + 9 - 6 - 6 = -3 \][/tex]

## Step 4: Write the resulting polynomial in standard form.
Combining all these results into a single polynomial, we get:
[tex]\[ -19g^4 - 5g^3 + 6g^2 + 29g - 3 \][/tex]

Thus, the sum of the given polynomial expressions is:
[tex]\[ -19g^4 - 5g^3 + 6g^2 + 29g - 3 \][/tex]
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