Problem 58 - College Algebra

Question

Factor using the formula for the sum or difference of tho cubes. $$ x^{3}+64 $$

Step-by-Step Solution

Answer

Factored form: (x+4)(x^2 - 4x + 16)

Step 1: Identify the values of 'a' and 'b'

Firstly, note that the expression can be rewritten using '^3' notation. It is x^3 + 4^3. So, 'a' is 'x' and 'b' is '4'.

Step 2: Apply the Formula

Now, apply the rule for the sum of cubes: a^3 + b^3 = (a+b)(a^2 - ab + b^2). Substitute 'x' into 'a' and '4' into 'b' for the expression x^3 + 4^3. So, x^3 + 4^3 = (x+4)(x^2 - 4x + 16).

Step 3: Final Simplified Form

The expression is now completely factored and can't be simplified any further, so the solution is (x+4)(x^2 - 4x + 16).

Key Concepts

Sum of Cubes

When dealing with polynomial expressions, you might come across forms like \(a^3 + b^3\). This form is known as the \'sum of cubes\'. Factoring these expressions follows a specific pattern. The sum of cubes is factored using the formula: - \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)This pattern might seem a bit complex at first, but once you break it down, it becomes clearer.

• The first term in the factorization is a simple sum of \(a\) and \(b\), which is \( (a+b) \).
• The second part \((a^2 - ab + b^2)\) is a trinomial. This is sometimes called the "remainder" when factoring the sum of cubes.

With practice, using this formula becomes intuitive, especially because it consistently applies to all compatible expressions. When looking at \(x^3 + 64\), \(x\) is \(a\) and \(4^3\) is \(b^3\). Plugging these into the formula makes factoring straightforward.

Difference of Cubes

Another concept to understand when factoring polynomials is the difference of cubes, evident in expressions like \(a^3 - b^3\). While similar to the sum of cubes, there is a crucial change in the middle sign of the formula:- \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)This difference formula shares the same basic structure with its \'sum\' counterpart:

• The first factor \((a-b)\) contains the difference between \(a\) and \(b\).
• The trinomial factor \((a^2 + ab + b^2)\) follows with a positive middle term \(ab\), contrasting the sum of cubes.

The change of signs is crucial when differentiating between these two formulas. Having a good grasp of these two patterns will make polynomial factoring much smoother and more logical.

Polynomial Expressions

Polynomial expressions serve as the building blocks for many algebraic concepts, including factoring with sums and differences of cubes. They are mathematical expressions that consist of variables raised to whole number powers and coefficients.Polynomials can take many forms, such as linear, quadratic, or cubic. Their degree is determined by the highest power of the variable. In the step-by-step solution, we're dealing specifically with a cubic polynomial, \(x^3+64\). Here, the expression is neatly set up for applying the sum of cubes formula.Factoring polynomial expressions, like the one in our exercise, requires recognizing these forms and applying appropriate strategies. It's helpful to:

• Identify variables and constants that fit known patterns (sums or differences of cubes, squares, etc.).
• Use formulas and identities to simplify or factor them completely.

Mastering polynomials and their factorization helps solve various algebraic problems. The ability to factor efficiently opens doors to solving equations, simplifying expressions, and solving real-world mathematical challenges.