Factor each expression using the greatest common factor. 8x – 28; 12x + 15
The Correct Answer and Explanation is:
To factor both expressions using the greatest common factor (GCF), we need to find the largest factor that divides all the terms in the expression.
Expression 1: 8x−288x – 288x−28
- Find the GCF:
- The coefficients of the terms are 8 and 28.
- The GCF of 8 and 28 is 4 because 4 is the largest number that divides both 8 and 28 evenly.
- Factor out the GCF:
- We can factor out 4 from both terms: 8x−28=4(2x−7)8x – 28 = 4(2x – 7)8x−28=4(2x−7)
- This is the factored form of the expression.
Expression 2: 12x+1512x + 1512x+15
- Find the GCF:
- The coefficients of the terms are 12 and 15.
- The GCF of 12 and 15 is 3 because 3 is the largest number that divides both 12 and 15 evenly.
- Factor out the GCF:
- We can factor out 3 from both terms: 12x+15=3(4x+5)12x + 15 = 3(4x + 5)12x+15=3(4x+5)
- This is the factored form of the expression.
Explanation:
When factoring expressions using the GCF, the first step is to identify the largest number that divides all the terms of the polynomial. Once the GCF is found, you can factor it out from each term in the expression, leaving a simpler expression in parentheses.
In both cases, the GCF is the key to simplifying the expression. For example, in the expression 8x−288x – 288x−28, after factoring out 4, the remaining expression inside the parentheses is 2x−72x – 72x−7, which cannot be simplified further. Similarly, in 12x+1512x + 1512x+15, factoring out 3 leaves 4x+54x + 54x+5, which is also in its simplest form.
Thus, both expressions are now factored using their GCFs:
- 8x−28=4(2x−7)8x – 28 = 4(2x – 7)8x−28=4(2x−7)
- 12x+15=3(4x+5)12x + 15 = 3(4x + 5)12x+15=3(4x+5)
