Fully factor the expression [tex]$4 x^2+6 x$[/tex]
Answer (1)
The problem requires factoring the expression 4 x 2 + 6 x .
Find the greatest common factor (GCF) of the coefficients 4 and 6, which is 2.
Find the GCF of the variables x 2 and x , which is x .
Factor out the GCF 2 x from the expression, resulting in 2 x ( 2 x + 3 ) .
The fully factored expression is 2 x ( 2 x + 3 ) .
Explanation
Understanding the Problem We are asked to fully factor the expression 4 x 2 + 6 x . This means we want to find the greatest common factor (GCF) of the terms and factor it out.
Finding the GCF of the Coefficients First, let's find the GCF of the coefficients, 4 and 6. The factors of 4 are 1, 2, and 4. The factors of 6 are 1, 2, 3, and 6. The greatest common factor of 4 and 6 is 2.
Finding the GCF of the Variables Next, let's find the GCF of the variable terms, x 2 and x . The GCF is x since it is the highest power of x that divides both terms.
Factoring out the GCF Now, we factor out the GCF, which is 2 x , from the expression 4 x 2 + 6 x . We divide each term by 2 x :
2 x 4 x 2 = 2 x 2 x 6 x = 3
Writing the Factored Expression So, the factored expression is 2 x ( 2 x + 3 ) .
Selecting the Correct Option Comparing this with the given multiple-choice options, we see that the correct answer is 2 x ( 2 x + 3 ) .
Examples
Factoring is a fundamental skill in algebra and is used in many real-world applications. For example, if you are designing a rectangular garden with an area represented by the expression 4 x 2 + 6 x , factoring it into 2 x ( 2 x + 3 ) can help you determine possible dimensions for the garden. One side could be 2 x and the other 2 x + 3 . This skill is also crucial in simplifying complex equations and solving problems in physics, engineering, and economics.
