Problem 35 - Algebra 1: Concepts and Skills

Question

Factor the expression. \(m^{3}-125\)

Step-by-Step Solution

Answer

The factored form of the given expression \(m^{3}-125\) is \((m - 5)(m^{2} + 5m + 25)\).

Step 1: Identify 'a' and 'b'

Start off by identifying 'a' and 'b' in the equation \(m^{3}-125\). Here, \(a = m\) since \(m = (m)^{1/3}\). And \(b = 5\) as \(125 = (5)^3\). The given equation can be rewritten as \(a^{3} - b^{3}\).

Step 2: Apply the Difference of Cubes Formula

The next step is to apply the difference of cubes formula: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Substitute 'a' and 'b' into the formula: \((m - 5)((m)^2 + 5m + 25)\).

Step 3: Final answer

The simplified form is \((m - 5)(m^{2} + 5m + 25)\), where the first part is \((m - 5)\) and the second part is \((m^{2} + 5m + 25)\). Notice how the \(m^{2} + 5m + 25\) cannot be factored any further as it doesn't have any real roots. The expression \(m^{3}-125\) is thus factored as \((m - 5)(m^{2} + 5m + 25)\).

Key Concepts

Difference of Cubes

When you encounter a difference of cubes problem, it looks like this: \[a^3 - b^3\]. This type of expression involves subtracting one cube from another.
To solve these problems, we use the difference of cubes factoring formula:

• \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)

This formula is handy because it breaks down a seemingly complex expression into a simpler, easy-to-handle form.
Applying this formula requires you to identify two crucial parts: 'a' and 'b'. These are the cube roots of the terms in the original expression.
For example, in \[m^3 - 125\], we see that \(a = m\) and \(b = 5\) since 125 is \(5^3\). After identifying these, substitute them into the formula to get the desired factored form.
Remember, using this formula helps you easily tackle cube-related algebra problems!

Polynomials

Polynomials are expressions made up of variables and constants.
They involve operations like addition, subtraction, multiplication, and exponentiation by a non-negative integer.
In simple terms, they look like this: \[a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\], where \(a_n\) to \(a_0\) are constants, and \(x\) is a variable.
When you factor a polynomial like \[m^3 - 125\], you're breaking it into simpler polynomials that can be multi-plied together to get the original expression.
Factoring helps to simplify equations, solve algebraic expressions, and find the roots of equations, among other things.

• Factoring polynomials reduces complex problems to simpler ones.
• Aids in finding solutions to polynomial equations.

This makes understanding polynomials crucial for algebraic operations!

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They don't have an equals sign like equations do. Instead, they're more like mathematical sentences that need interpretation or simplification.
For instance, \[m^3 - 125\] is an algebraic expression showcasing subtraction in polynomial form.

• These expressions follow algebraic rules for simplification and manipulation.
• They're used in solving equations, modeling real-world situations, and more.

Understanding expressions is key to mastering Algebra. Breaking them down into factored forms like \[(m - 5)(m^2 + 5m + 25)\], as seen here, simplifies solving or analyzing them.
This simplification helps in viewing the individual parts that compose the full expression, leading to deeper insight and easier resolution.