Answer :
To solve the problem of subtracting the polynomials [tex]\( (7x^7 - 5x^6 + 2x^5 - 2) \)[/tex] and [tex]\( (2x^7 + 7x^6 + 5x^5 - 7) \)[/tex], we perform the subtraction term by term. Here’s a step-by-step explanation:
1. Identify the Terms:
- First Polynomial: [tex]\( 7x^7 - 5x^6 + 2x^5 - 2 \)[/tex]
- Second Polynomial: [tex]\( 2x^7 + 7x^6 + 5x^5 - 7 \)[/tex]
2. Subtract Corresponding Terms:
- For [tex]\( x^7 \)[/tex] term:
- [tex]\( 7x^7 - 2x^7 = (7 - 2)x^7 = 5x^7 \)[/tex]
- For [tex]\( x^6 \)[/tex] term:
- [tex]\(-5x^6 - 7x^6 = (-5 - 7)x^6 = -12x^6 \)[/tex]
- For [tex]\( x^5 \)[/tex] term:
- [tex]\( 2x^5 - 5x^5 = (2 - 5)x^5 = -3x^5 \)[/tex]
- For the constant term:
- [tex]\(-2 - (-7) = -2 + 7 = 5\)[/tex]
3. Write the Resulting Polynomial:
By combining all these calculations, the resulting polynomial after the subtraction is:
[tex]\[
5x^7 - 12x^6 - 3x^5 + 5
\][/tex]
This expression represents the polynomial you get after performing the indicated operation.
1. Identify the Terms:
- First Polynomial: [tex]\( 7x^7 - 5x^6 + 2x^5 - 2 \)[/tex]
- Second Polynomial: [tex]\( 2x^7 + 7x^6 + 5x^5 - 7 \)[/tex]
2. Subtract Corresponding Terms:
- For [tex]\( x^7 \)[/tex] term:
- [tex]\( 7x^7 - 2x^7 = (7 - 2)x^7 = 5x^7 \)[/tex]
- For [tex]\( x^6 \)[/tex] term:
- [tex]\(-5x^6 - 7x^6 = (-5 - 7)x^6 = -12x^6 \)[/tex]
- For [tex]\( x^5 \)[/tex] term:
- [tex]\( 2x^5 - 5x^5 = (2 - 5)x^5 = -3x^5 \)[/tex]
- For the constant term:
- [tex]\(-2 - (-7) = -2 + 7 = 5\)[/tex]
3. Write the Resulting Polynomial:
By combining all these calculations, the resulting polynomial after the subtraction is:
[tex]\[
5x^7 - 12x^6 - 3x^5 + 5
\][/tex]
This expression represents the polynomial you get after performing the indicated operation.
