Problem 29 - Elementary Algebra

Question

For the following problems, factor the binomials. $$ 12 a^{2}-75 $$

Step-by-Step Solution

Answer

The factored form of the expression is 3(2a + 5)(2a - 5).

Step 1: Identify the GCF

The given expression is: $$ 12 a^{2}-75 $$ Let's find the greatest common factor (GCF) of 12 and 75. The factors of 12 are: (1, 12), (2, 6), (3, 4). The factors of 75 are: (1, 75), (3, 25), (5, 15). The greatest common factor between the two lists is 3. Now let's factor out the GCF (3) from the expression:

Step 2: Factor out the GCF

We found that 3 is the GCF of 12 and 75, so we can factor out 3 from the expression. The expression will now look like: $$ 3(4 a^{2} - 25) $$ Now within the parentheses, we have a difference of squares, since 4 and 25 are perfect squares.

Step 3: Identify difference of squares

The expression inside the parentheses looks like: $$ 4 a^{2} - 25 $$ It can be rewritten as a difference of squares as: $$ (2a)^{2} - (5)^{2} $$ We can use the formula: \((A^2 - B^2) = (A + B)(A - B)\) to factor this difference of squares.

Step 4: Apply the difference of squares formula

Now let's apply the difference of squares formula on the expression inside the parentheses: $$ (2a + 5)(2a - 5) $$ Don't forget to include the GCF (3) we found earlier. So, the factored expression is:

Step 5: Final answer

The final factored expression will look like: $$ 3(2a + 5)(2a - 5) $$

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is an essential concept when learning to factor binomials. It is the highest number that can evenly divide two or more numbers. In our example, we need to identify the GCF of 12 and 75.

• Factors of 12 are 1, 2, 3, 4, 6, and 12.
• Factors of 75 are 1, 3, 5, 15, 25, and 75.

The greatest factor common to both 12 and 75 is 3.Why is finding the GCF important? It simplifies binomials by reducing complex terms. In this exercise, factoring out the GCF transforms the expression from \(12a^2 - 75\) to \(3(4a^2 - 25)\). Always look for the GCF first when factoring binomials; it makes the problem significantly more manageable.

Difference of Squares

The difference of squares is a powerful algebraic tool used to simplify expressions. It involves expressions that can be written as two squares subtracted from one another. In our example, after factoring out the GCF, we are left with \(4a^2 - 25\).Notice:- \(4a^2 = (2a)^2\) is a perfect square.- \(25 = (5)^2\) is also a perfect square.This forms what we call a 'difference of squares' scenario. The general formula for factoring a difference of squares is:\[A^2 - B^2 = (A + B)(A - B)\]Applying this formula, we convert \((2a)^2 - 5^2\) into \((2a + 5)(2a - 5)\). Recognizing the difference of squares is key in simplifying binomials effectively.

Perfect Squares

Perfect squares are numbers that can be expressed as the square of an integer. Recognizing them in binomials allows us to apply specific factoring techniques.In our expression \(4a^2 - 25\), both components are perfect squares:

• \(4a^2\) is \((2a)^2\)
• \(25\) is \(5^2\)

Identifying perfect squares helps to see whether the expression fits the difference of squares pattern. Factoring becomes straightforward once perfect squares are spotted, using the difference of squares formula. Being adept at identifying perfect squares, like recognizing \(4\) from \((2)^2\) or \(25\) from \(5^2\), ensures you can quickly factorize any eligible binomial expressions.