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Factoring polynomials involves breaking down a polynomial into simpler polynomials that, when multiplied together, give the original polynomial. This skill is crucial in algebra for simplifying expressions and solving equations more easily.
For example, with more complex polynomials, recognizing patterns such as the difference of squares, perfect square trinomials, or the sum and difference of cubes can simplify the factoring process significantly.
In the given exercise, we factored the polynomial \(x^3 + 64\) by recognizing that it is a sum of cubes.

The sum of cubes formula is a crucial tool in polynomial factoring. It helps in simplifying expressions of the form \(a^3 + b^3\).
The formula is given by:
\[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]
To use this formula:

• Identify the terms that are cubes. For instance, in \(x^3 + 64\), we see that \(x^3\) is \(a^3\) and 64 is \(4^3\).
• Match these terms to \(a^3\) and \(b^3\) where \(a = x\) and \(b = 4\).
• Substitute these values into the sum of cubes formula to get \((x + 4)(x^2 - 4x + 16)\).

This process decomposes the original polynomial into simpler factors.

An algebraic expression is a mathematical phrase that can include numbers, variables, and operations. Understanding and manipulating algebraic expressions is fundamental in algebra and essential for solving equations and word problems.
Let's take the polynomial \(x^3 + 64\). It's an algebraic expression because it consists of the variable \(x\) and the constant 64, connected through the operation of addition. By factoring the polynomial, we rewrite it in a different form, thus simplifying the expression.
In general, becoming comfortable with algebraic expressions involves recognizing patterns and applying the right formulas, like the sum of cubes, to make the expressions more manageable.