To determine the first and last terms of the expression \((12x^5+4x^4+9x^3−10x^2+15)−(24x^5+9x^3−7x^2+8)\), we first need to simplify the expression by distributing the negative sign and combining like terms.
Starting with the original expression:
\[ (12x^5 + 4x^4 + 9x^3 - 10x^2 + 15) - (24x^5 + 9x^3 - 7x^2 + 8) \]
Distribute the negative to each term in the second polynomial:
\[ = 12x^5 + 4x^4 + 9x^3 - 10x^2 + 15 - 24x^5 - 9x^3 + 7x^2 - 8 \]
Now, combine like terms:
- For \(x^5\) terms: \(12x^5 - 24x^5 = -12x^5\)
- For \(x^4\) terms: \(4x^4\) (there's no other \(x^4\) term)
- For \(x^3\) terms: \(9x^3 - 9x^3 = 0\)
- For \(x^2\) terms: \(-10x^2 + 7x^2 = -3x^2\)
- Constant terms: \(15 - 8 = 7\)
Combining these results gives:
\[ -12x^5 + 4x^4 - 3x^2 + 7 \]
Now, we identify the first and last terms of this simplified polynomial.
- First term (the term with the highest degree): \(-12x^5\)
- Last term (the constant): \(7\)
Thus, the first term is \(-12x^5\) and the last term is \(7\).
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