Factor the following expression completely. If the expression is not factorable, indicate that it is not factorable.
[tex]12 x^2+27[/tex]
Answer (2)
The expression 12 x 2 + 27 can be factored by first identifying the greatest common factor, which is 3. After factoring out 3, we are left with 3 ( 4 x 2 + 9 ) , where 4 x 2 + 9 cannot be factored further using real numbers. Therefore, the completely factored form is 3 ( 4 x 2 + 9 ) .
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Find the greatest common factor (GCF) of the coefficients 12 and 27, which is 3.
Factor out the GCF from the expression: 12 x 2 + 27 = 3 ( 4 x 2 + 9 ) .
Check if the remaining expression 4 x 2 + 9 can be factored further. Since it is a sum of squares, it cannot be factored further using real numbers.
The completely factored form of the expression is 3 ( 4 x 2 + 9 ) .
Explanation
Understanding the Problem We are asked to factor the expression 12 x 2 + 27 completely. This means we want to find the greatest common factor (GCF) of the terms and factor it out. Then we check if the remaining expression can be factored further.
Finding the GCF First, we find the GCF of the coefficients 12 and 27. The GCF of 12 and 27 is 3, since 12 = 3 × 4 and 27 = 3 × 9 .
Factoring out the GCF Now, we factor out the GCF, which is 3, from the expression: 12 x 2 + 27 = 3 ( 4 x 2 + 9 )
Checking for Further Factoring Next, we check if the expression 4 x 2 + 9 can be factored further. Notice that 4 x 2 is a perfect square, ( 2 x ) 2 , and 9 is also a perfect square, 3 2 . However, since it is a sum of squares, not a difference of squares, it cannot be factored further using real numbers. Therefore, the completely factored form of the expression is 3 ( 4 x 2 + 9 ) .
Final Answer The completely factored form of the expression 12 x 2 + 27 is 3 ( 4 x 2 + 9 ) .
Examples
Factoring expressions like 12 x 2 + 27 is useful in many areas of mathematics, such as solving equations, simplifying algebraic expressions, and analyzing functions. For example, if you were designing a rectangular garden with an area represented by 12 x 2 + 27 square feet, factoring the expression could help you determine possible dimensions for the garden in terms of x . Understanding how to factor such expressions allows for efficient problem-solving and provides insights into the underlying structure of mathematical relationships.
