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Answer :
To find the difference of the polynomials
[tex]$$
\left(12x^3 + 10x + 4\right) - \left(15x^3 + 7x^2 + 12x + 9\right),
$$[/tex]
we proceed as follows:
1. Write the expression by distributing the subtraction across the second polynomial:
[tex]$$
12x^3 + 10x + 4 - 15x^3 - 7x^2 - 12x - 9.
$$[/tex]
2. Group the like terms by their degree:
- For the [tex]$x^3$[/tex] terms: [tex]$$12x^3 - 15x^3.$$[/tex]
- For the [tex]$x^2$[/tex] terms: There is no [tex]$x^2$[/tex] term in the first polynomial, so it becomes: [tex]$$0 - 7x^2.$$[/tex]
- For the [tex]$x$[/tex] terms: [tex]$$10x - 12x.$$[/tex]
- For the constant terms: [tex]$$4 - 9.$$[/tex]
3. Subtract the coefficients:
- [tex]$x^3$[/tex] terms:
[tex]$$
12 - 15 = -3 \quad \text{so the term is } -3x^3.
$$[/tex]
- [tex]$x^2$[/tex] terms:
[tex]$$
0 - 7 = -7 \quad \text{so the term is } -7x^2.
$$[/tex]
- [tex]$x$[/tex] terms:
[tex]$$
10 - 12 = -2 \quad \text{so the term is } -2x.
$$[/tex]
- Constant terms:
[tex]$$
4 - 9 = -5 \quad \text{so the constant is } -5.
$$[/tex]
4. Combine the results to form the resulting polynomial:
[tex]$$
-3x^3 - 7x^2 - 2x - 5.
$$[/tex]
Among the given options, this expression corresponds to option D.
[tex]$$
\left(12x^3 + 10x + 4\right) - \left(15x^3 + 7x^2 + 12x + 9\right),
$$[/tex]
we proceed as follows:
1. Write the expression by distributing the subtraction across the second polynomial:
[tex]$$
12x^3 + 10x + 4 - 15x^3 - 7x^2 - 12x - 9.
$$[/tex]
2. Group the like terms by their degree:
- For the [tex]$x^3$[/tex] terms: [tex]$$12x^3 - 15x^3.$$[/tex]
- For the [tex]$x^2$[/tex] terms: There is no [tex]$x^2$[/tex] term in the first polynomial, so it becomes: [tex]$$0 - 7x^2.$$[/tex]
- For the [tex]$x$[/tex] terms: [tex]$$10x - 12x.$$[/tex]
- For the constant terms: [tex]$$4 - 9.$$[/tex]
3. Subtract the coefficients:
- [tex]$x^3$[/tex] terms:
[tex]$$
12 - 15 = -3 \quad \text{so the term is } -3x^3.
$$[/tex]
- [tex]$x^2$[/tex] terms:
[tex]$$
0 - 7 = -7 \quad \text{so the term is } -7x^2.
$$[/tex]
- [tex]$x$[/tex] terms:
[tex]$$
10 - 12 = -2 \quad \text{so the term is } -2x.
$$[/tex]
- Constant terms:
[tex]$$
4 - 9 = -5 \quad \text{so the constant is } -5.
$$[/tex]
4. Combine the results to form the resulting polynomial:
[tex]$$
-3x^3 - 7x^2 - 2x - 5.
$$[/tex]
Among the given options, this expression corresponds to option D.
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