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Answer :
Sure! Let's find the difference of the given polynomials step-by-step.
We are given the polynomials:
[tex]\[ P(x) = 7x^9 + 9x^5 + 15 \][/tex]
and
[tex]\[ Q(x) = 10x^9 + 8x^5 + 12 \][/tex]
We need to find the difference [tex]\( P(x) - Q(x) \)[/tex].
### Step 1: Subtract the coefficients of [tex]\( x^9 \)[/tex]
The coefficient of [tex]\( x^9 \)[/tex] in [tex]\( P(x) \)[/tex] is 7.
The coefficient of [tex]\( x^9 \)[/tex] in [tex]\( Q(x) \)[/tex] is 10.
So, the difference for [tex]\( x^9 \)[/tex] is:
[tex]\[ 7 - 10 = -3 \][/tex]
### Step 2: Subtract the coefficients of [tex]\( x^5 \)[/tex]
The coefficient of [tex]\( x^5 \)[/tex] in [tex]\( P(x) \)[/tex] is 9.
The coefficient of [tex]\( x^5 \)[/tex] in [tex]\( Q(x) \)[/tex] is 8.
So, the difference for [tex]\( x^5 \)[/tex] is:
[tex]\[ 9 - 8 = 1 \][/tex]
### Step 3: Subtract the constant terms
The constant term in [tex]\( P(x) \)[/tex] is 15.
The constant term in [tex]\( Q(x) \)[/tex] is 12.
So, the difference for the constant terms is:
[tex]\[ 15 - 12 = 3 \][/tex]
### Putting it all together
Combining the differences from the above steps, we get:
[tex]\[ -3x^9 + x^5 + 3 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{-3x^9 + x^5 + 3} \][/tex]
This matches option D from the given choices:
D. [tex]\(-3x^9 + x^5 + 3\)[/tex]
Therefore, the final answer is:
[tex]\[ \boxed{D} \][/tex]
We are given the polynomials:
[tex]\[ P(x) = 7x^9 + 9x^5 + 15 \][/tex]
and
[tex]\[ Q(x) = 10x^9 + 8x^5 + 12 \][/tex]
We need to find the difference [tex]\( P(x) - Q(x) \)[/tex].
### Step 1: Subtract the coefficients of [tex]\( x^9 \)[/tex]
The coefficient of [tex]\( x^9 \)[/tex] in [tex]\( P(x) \)[/tex] is 7.
The coefficient of [tex]\( x^9 \)[/tex] in [tex]\( Q(x) \)[/tex] is 10.
So, the difference for [tex]\( x^9 \)[/tex] is:
[tex]\[ 7 - 10 = -3 \][/tex]
### Step 2: Subtract the coefficients of [tex]\( x^5 \)[/tex]
The coefficient of [tex]\( x^5 \)[/tex] in [tex]\( P(x) \)[/tex] is 9.
The coefficient of [tex]\( x^5 \)[/tex] in [tex]\( Q(x) \)[/tex] is 8.
So, the difference for [tex]\( x^5 \)[/tex] is:
[tex]\[ 9 - 8 = 1 \][/tex]
### Step 3: Subtract the constant terms
The constant term in [tex]\( P(x) \)[/tex] is 15.
The constant term in [tex]\( Q(x) \)[/tex] is 12.
So, the difference for the constant terms is:
[tex]\[ 15 - 12 = 3 \][/tex]
### Putting it all together
Combining the differences from the above steps, we get:
[tex]\[ -3x^9 + x^5 + 3 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{-3x^9 + x^5 + 3} \][/tex]
This matches option D from the given choices:
D. [tex]\(-3x^9 + x^5 + 3\)[/tex]
Therefore, the final answer is:
[tex]\[ \boxed{D} \][/tex]
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