Factor the expression completely. 48x-42
The correct answer and explanation is:
To factor the expression 48x−4248x – 42, the first step is to find the greatest common factor (GCF) of the two terms.
Step 1: Find the GCF
The two terms in the expression are 48x48x and −42-42. To find the GCF, look at the coefficients of the terms (48 and 42) and the variable part of 48x48x.
- The prime factorization of 48 is 48=24×348 = 2^4 \times 3.
- The prime factorization of 42 is 42=2×3×742 = 2 \times 3 \times 7.
The common factors between 48 and 42 are 2 and 3. So, the GCF of 48 and 42 is 2×3=62 \times 3 = 6.
Step 2: Factor out the GCF
Now that we know the GCF is 6, factor 6 out of the expression: 48x−42=6(8x−7)48x – 42 = 6(8x – 7)
Step 3: Check if further factoring is possible
Now, examine the factored expression 6(8x−7)6(8x – 7). The term 8x−78x – 7 is a binomial, and it cannot be factored further because it does not have any common factors or special factoring patterns.
Final Factored Form
Thus, the completely factored form of the expression is: 6(8x−7)6(8x – 7)
Conclusion
By finding the greatest common factor of the two terms, which is 6, and factoring it out, the expression 48x−4248x – 42 becomes 6(8x−7)6(8x – 7). This is the simplest and fully factored form of the expression. Factoring is useful because it simplifies expressions and is often a key step in solving equations or simplifying algebraic problems.