Problem 310 - Elementary Algebra

Question

Factor completely. \(3 v^{4}-768\)

Step-by-Step Solution

Answer

\[3(v - 4)(v + 4)(v^2 + 16)\]

Step 1: Identify and factor out the greatest common factor (GCF)

First, observe that both terms, 3 and -768, share a common factor of 3. Factor out this greatest common factor from the expression: \[3 v^4 - 768 = 3 ( v^4 - 256 )\]

Step 2: Recognize the difference of squares

Within the parentheses, the expression \(v^4 - 256\) is a difference of squares, as it can be written as \(v^4 - 16^2\). The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\).

Step 3: Apply the difference of squares formula

Using the formula, factor \(v^4 - 256\) as follows: \[v^4 - 16^2 = (v^2 - 16)(v^2 + 16)\]So, the expression becomes: \[3(v^2 - 16)(v^2 + 16)\]

Step 4: Factor further if possible

Notice that \(v^2 - 16\) can be factored further, as it is also a difference of squares: \[v^2 - 16 = (v - 4)(v + 4)\]So the final factored form of the expression is: \[3(v - 4)(v + 4)(v^2 + 16)\]

Key Concepts

difference of squares

Understanding the 'difference of squares' is essential in polynomial factorization. A difference of squares is a specific type of quadratic expression that takes the form: \[ a^2 - b^2 \] Where both terms are perfect squares. This can be factored into: \[ (a - b)(a + b) \] For example, in our exercise: \[ v^4 - 256 \] We can rewrite 256 as \(16^2\), identifying it as a difference of squares: \[ v^4 - 16^2 = (v^2 - 16)(v^2 + 16) \] This special structure simplifies complex polynomials into easier factors.

greatest common factor

The 'greatest common factor' (GCF) is the largest factor shared by all terms in an algebraic expression. To find the GCF, look for the highest common divisor among the coefficients and variables. In our exercise, we identify the GCF of 3 and -768, which is 3. Factoring out the GCF transforms the original polynomial: \[ 3 v^4 - 768 = 3 ( v^4 - 256 ) \] This simplifies the expression, making it easier to handle subsequent steps.

polynomial factorization

Polynomial factorization involves breaking down a polynomial into simpler 'factor' polynomials that, when multiplied together, yield the original polynomial. Here, we factored: \[ 3 v^4 - 768 \] Step-by-step, we:

• Factored out the GCF: 3
• Identified and applied the difference of squares: \( v^4 - 256 \)
• Factored further difference of squares: \( v^2 - 16 \)

So, the final factored form is: \[ 3(v - 4)(v + 4)(v^2 + 16) \]

algebraic expressions

Algebraic expressions contain numbers, variables, and operations. Factoring these expressions simplifies solving equations or evaluating functions. Consider: \[ 3 v^4 - 768 \] We can rewrite it in more manageable forms using algebraic properties such as factoring out the GCF and applying special formulas like the difference of squares. This process streamlines problem-solving and reveals deeper insights into the structure of the expression.