Sure, let's break down the expression [tex]\( 78 A^6 b^4 - 54 A^4 b^2 + 36 a^2 b \)[/tex] step-by-step to analyze its components.
1. Analyzing the terms individually:
- The first term is [tex]\( 78 A^6 b^4 \)[/tex], which means:
- [tex]\( A \)[/tex] is raised to the power of 6.
- [tex]\( b \)[/tex] is raised to the power of 4.
- This term is then multiplied by 78.
- The second term is [tex]\( -54 A^4 b^2 \)[/tex], which means:
- [tex]\( A \)[/tex] is raised to the power of 4.
- [tex]\( b \)[/tex] is raised to the power of 2.
- This term is then multiplied by -54.
- The third term is [tex]\( 36 a^2 b \)[/tex], which means:
- [tex]\( a \)[/tex] is raised to the power of 2.
- [tex]\( b \)[/tex] is included as a factor.
- This term is then multiplied by 36.
2. Understanding subtraction and addition:
- The expression consists of addition and subtraction operations among these three terms. Specifically, it subtracts [tex]\( 54 A^4 b^2 \)[/tex] from [tex]\( 78 A^6 b^4 \)[/tex] and then adds [tex]\( 36 a^2 b \)[/tex].
3. Putting it all together:
- The expression [tex]\( 78 A^6 b^4 - 54 A^4 b^2 + 36 a^2 b \)[/tex] can be seen as a polynomial in terms of [tex]\( A \)[/tex] and [tex]\( b \)[/tex].
- Each term has individual constant coefficients (78, -54, and 36) and different powers of the variables [tex]\( A \)[/tex] and [tex]\( b \)[/tex].
4. Grouping similar powers:
- The first two terms, [tex]\( 78 A^6 b^4 \)[/tex] and [tex]\( -54 A^4 b^2 \)[/tex], involve the variable [tex]\( A \)[/tex] raised to different powers and the variable [tex]\( b \)[/tex] raised to different powers.
- The last term, [tex]\( 36 a^2 b \)[/tex], involves a different variable [tex]\( a \)[/tex], indicating that the variables [tex]\( A \)[/tex] and [tex]\( a \)[/tex] are distinct and should be treated accordingly.
Therefore, the expression [tex]\( 78 A^6 b^4 - 54 A^4 b^2 + 36 a^2 b \)[/tex] is a combination of polynomial terms featuring variables [tex]\( A \)[/tex], [tex]\( a \)[/tex], and [tex]\( b \)[/tex] raised to various powers and multiplied by corresponding constants.
This concludes the detailed step-by-step breakdown of the given expression.
