Factor 63y+81z.
The correct answer and explanation is:
To factor the expression 63y+81z63y + 81z, we need to identify the greatest common factor (GCF) of the two terms.
- First, observe the numerical coefficients of the two terms: 63 and 81.
- The prime factorization of 63 is 63=32×763 = 3^2 \times 7.
- The prime factorization of 81 is 81=3481 = 3^4.
- The GCF of 63 and 81 is 32=93^2 = 9, since 9 is the highest power of 3 that divides both 63 and 81.
- Next, factor out the GCF, which is 9: 63y+81z=9(7y+9z)63y + 81z = 9(7y + 9z)
- Now the expression is written as the product of 9 and the binomial (7y+9z)(7y + 9z).
Explanation:
Factoring involves finding the greatest common factor (GCF) of the terms in the expression. In this case, 63 and 81 share a GCF of 9. Once we factor out 9 from both terms, the remaining expression is 7y+9z7y + 9z.
Factoring out the GCF helps simplify the expression and makes it easier to work with, especially when solving equations or simplifying further. It is an essential skill in algebra, where recognizing common factors can often lead to solutions or further simplifications in various mathematical problems. This method is helpful in solving real-world problems where expressions need to be broken down into simpler forms for easier computation or interpretation.