Let's find the expression for [tex]\(3C - 2D\)[/tex] in standard form given [tex]\(C = x^2 + 7x + 5\)[/tex] and [tex]\(D = x^2 - 8\)[/tex].
1. Calculate [tex]\(3C\)[/tex]:
[tex]\[
C = x^2 + 7x + 5
\][/tex]
Multiplying this expression by 3:
[tex]\[
3C = 3(x^2 + 7x + 5)
\][/tex]
Distribute the 3 across each term in the parentheses:
[tex]\[
3C = 3x^2 + 21x + 15
\][/tex]
2. Calculate [tex]\(2D\)[/tex]:
[tex]\[
D = x^2 - 8
\][/tex]
Multiplying this expression by 2:
[tex]\[
2D = 2(x^2 - 8)
\][/tex]
Distribute the 2 across each term in the parentheses:
[tex]\[
2D = 2x^2 - 16
\][/tex]
3. Subtract [tex]\(2D\)[/tex] from [tex]\(3C\)[/tex]:
[tex]\[
3C - 2D = (3x^2 + 21x + 15) - (2x^2 - 16)
\][/tex]
Distribute the subtraction across each term of [tex]\(2D\)[/tex]:
[tex]\[
3C - 2D = 3x^2 + 21x + 15 - 2x^2 + 16
\][/tex]
4. Combine like terms:
Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
3x^2 - 2x^2 = x^2
\][/tex]
Combine the [tex]\(x\)[/tex] terms (there is only one [tex]\(21x\)[/tex]):
[tex]\[
21x
\][/tex]
Combine the constant terms:
[tex]\[
15 + 16 = 31
\][/tex]
Putting it all together, we get:
[tex]\[
3C - 2D = x^2 + 21x + 31
\][/tex]
The expression in standard form for [tex]\(3C - 2D\)[/tex] is:
[tex]\[
x^2 + 21x + 31
\][/tex]
