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Answer :
Sure! Let's solve the given problem step-by-step to collect like terms and write the expression in descending order of powers of [tex]\( x \)[/tex].
The given expressions are:
[tex]\[ (4 - 7x^7 + 5x^9 + 5x^8) + (9x^8 - 9x^7 - 8 + 8x^9) \][/tex]
### Step 1: Write down the expressions.
First, let's write down both expressions clearly:
[tex]\[ 4 - 7x^7 + 5x^9 + 5x^8 \][/tex]
[tex]\[ 9x^8 - 9x^7 - 8 + 8x^9 \][/tex]
### Step 2: Combine the expressions.
Now, add the two expressions together:
[tex]\[ (4 - 7x^7 + 5x^9 + 5x^8) + (9x^8 - 9x^7 - 8 + 8x^9) \][/tex]
### Step 3: Group like terms.
Group the terms with [tex]\( x^9 \)[/tex], [tex]\( x^8 \)[/tex], [tex]\( x^7 \)[/tex], and constants separately:
- [tex]\( x^9 \)[/tex] terms: [tex]\( 5x^9 + 8x^9 \)[/tex]
- [tex]\( x^8 \)[/tex] terms: [tex]\( 5x^8 + 9x^8 \)[/tex]
- [tex]\( x^7 \)[/tex] terms: [tex]\( -7x^7 - 9x^7 \)[/tex]
- Constant terms: [tex]\( 4 - 8 \)[/tex]
### Step 4: Add the like terms.
Add the coefficients of the like terms:
- For [tex]\( x^9 \)[/tex] terms: [tex]\( 5 + 8 = 13 \)[/tex], so [tex]\( 5x^9 + 8x^9 = 13x^9 \)[/tex]
- For [tex]\( x^8 \)[/tex] terms: [tex]\( 5 + 9 = 14 \)[/tex], so [tex]\( 5x^8 + 9x^8 = 14x^8 \)[/tex]
- For [tex]\( x^7 \)[/tex] terms: [tex]\( -7 - 9 = -16 \)[/tex], so [tex]\( -7x^7 - 9x^7 = -16x^7 \)[/tex]
- For the constants: [tex]\( 4 - 8 = -4 \)[/tex]
### Step 5: Write the final expression in descending order.
Combine all the simplified terms:
[tex]\[ 13x^9 + 14x^8 - 16x^7 - 4 \][/tex]
So, the expression in descending order and with like terms collected is:
[tex]\[ 13x^9 + 14x^8 - 16x^7 - 4 \][/tex]
This is the final simplified expression.
The given expressions are:
[tex]\[ (4 - 7x^7 + 5x^9 + 5x^8) + (9x^8 - 9x^7 - 8 + 8x^9) \][/tex]
### Step 1: Write down the expressions.
First, let's write down both expressions clearly:
[tex]\[ 4 - 7x^7 + 5x^9 + 5x^8 \][/tex]
[tex]\[ 9x^8 - 9x^7 - 8 + 8x^9 \][/tex]
### Step 2: Combine the expressions.
Now, add the two expressions together:
[tex]\[ (4 - 7x^7 + 5x^9 + 5x^8) + (9x^8 - 9x^7 - 8 + 8x^9) \][/tex]
### Step 3: Group like terms.
Group the terms with [tex]\( x^9 \)[/tex], [tex]\( x^8 \)[/tex], [tex]\( x^7 \)[/tex], and constants separately:
- [tex]\( x^9 \)[/tex] terms: [tex]\( 5x^9 + 8x^9 \)[/tex]
- [tex]\( x^8 \)[/tex] terms: [tex]\( 5x^8 + 9x^8 \)[/tex]
- [tex]\( x^7 \)[/tex] terms: [tex]\( -7x^7 - 9x^7 \)[/tex]
- Constant terms: [tex]\( 4 - 8 \)[/tex]
### Step 4: Add the like terms.
Add the coefficients of the like terms:
- For [tex]\( x^9 \)[/tex] terms: [tex]\( 5 + 8 = 13 \)[/tex], so [tex]\( 5x^9 + 8x^9 = 13x^9 \)[/tex]
- For [tex]\( x^8 \)[/tex] terms: [tex]\( 5 + 9 = 14 \)[/tex], so [tex]\( 5x^8 + 9x^8 = 14x^8 \)[/tex]
- For [tex]\( x^7 \)[/tex] terms: [tex]\( -7 - 9 = -16 \)[/tex], so [tex]\( -7x^7 - 9x^7 = -16x^7 \)[/tex]
- For the constants: [tex]\( 4 - 8 = -4 \)[/tex]
### Step 5: Write the final expression in descending order.
Combine all the simplified terms:
[tex]\[ 13x^9 + 14x^8 - 16x^7 - 4 \][/tex]
So, the expression in descending order and with like terms collected is:
[tex]\[ 13x^9 + 14x^8 - 16x^7 - 4 \][/tex]
This is the final simplified expression.
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