Let's analyze and work through the given expression step-by-step.
### a) Writing the Expression in Descending Order of Powers of [tex]\( x \)[/tex]
The given expression is:
[tex]\[
8x^4 + 4x - 3x^2 - 12 + x^3
\][/tex]
To write this expression in descending order of powers of [tex]\( x \)[/tex], we need to arrange the terms starting with the highest power of [tex]\( x \)[/tex] and descending to the lowest power. Identifying each term's power of [tex]\( x \)[/tex], we get:
- [tex]\( 8x^4 \)[/tex] has the highest power, [tex]\( x^4 \)[/tex].
- [tex]\( x^3 \)[/tex] has the next highest power, [tex]\( x^3 \)[/tex].
- [tex]\( -3x^2 \)[/tex] follows with [tex]\( x^2 \)[/tex].
- [tex]\( 4x \)[/tex] follows with [tex]\( x^1 \)[/tex].
- Lastly, [tex]\( -12 \)[/tex] is the constant term, having [tex]\( x^0 \)[/tex].
Arranging them in descending order, the expression becomes:
[tex]\[
8x^4 + x^3 - 3x^2 + 4x - 12
\][/tex]
### b) Counting the Number of Terms in the Expression
A term in this context is any part of the expression separated by a plus (+) or minus (-) sign. Therefore, we count each distinct term in the rearranged expression:
[tex]\[
8x^4 + x^3 - 3x^2 + 4x - 12
\][/tex]
Here we have:
1. [tex]\( 8x^4 \)[/tex]
2. [tex]\( x^3 \)[/tex]
3. [tex]\( -3x^2 \)[/tex]
4. [tex]\( 4x \)[/tex]
5. [tex]\( -12 \)[/tex]
Thus, there are a total of 5 terms in the expression.
### Conclusion
a) The expression in descending order of powers of [tex]\( x \)[/tex] is:
[tex]\[
8x^4 + x^3 - 3x^2 + 4x - 12
\][/tex]
b) The number of terms in the expression is:
[tex]\[
5
\][/tex]
