Let's solve the expression step by step to find the first and last terms of the resultant expression:
We start by subtracting two polynomial expressions:
[tex]\[ \left(12 x^5+4 x^4+9 x^3-10 x^2+15\right)-\left(24 x^5+9 x^3-7 x^2+8\right) \][/tex]
1. Identify and Align Coefficients:
We align the coefficients of each corresponding power of [tex]\( x \)[/tex]:
- [tex]\( x^5 \)[/tex]: [tex]\( 12x^5 - 24x^5 \)[/tex]
- [tex]\( x^4 \)[/tex]: [tex]\( 4x^4 - 0x^4 \)[/tex] (no [tex]\( x^4 \)[/tex] term in the second expression, so it's effectively 0)
- [tex]\( x^3 \)[/tex]: [tex]\( 9x^3 - 9x^3 \)[/tex]
- [tex]\( x^2 \)[/tex]: [tex]\( -10x^2 - (-7x^2) \)[/tex]
- Constant term: [tex]\( 15 - 8 \)[/tex]
2. Perform the Subtraction for Each Term:
Carry out the subtraction for each corresponding term:
- The coefficient for [tex]\( x^5 \)[/tex]: [tex]\( 12 - 24 = -12 \)[/tex]
- The coefficient for [tex]\( x^4 \)[/tex]: [tex]\( 4 - 0 = 4 \)[/tex]
- The coefficient for [tex]\( x^3 \)[/tex]: [tex]\( 9 - 9 = 0 \)[/tex]
- The coefficient for [tex]\( x^2 \)[/tex]: [tex]\( -10 - (-7) = -10 + 7 = -3 \)[/tex]
- The constant term: [tex]\( 15 - 8 = 7 \)[/tex]
3. Form the Resulting Expression:
The resulting polynomial expression after subtracting is:
[tex]\[ -12x^5 + 4x^4 + 0x^3 - 3x^2 + 7 \][/tex]
From this expression, we can determine:
- The first (leading) term is the term with the highest power of [tex]\( x \)[/tex], which is [tex]\( -12x^5 \)[/tex]. The coefficient here is -12.
- The last term is the constant term, which is 7.
Therefore, the correct answer is:
- The first term is -12.
- The last term is 7.
Based on the given options, the correct determination of the first and last terms is:
"The first term is -12, and the last term is 7."
