Problem 298 - Elementary Algebra

Question

Factor completely. \(5 t^{3}-40\)

Step-by-Step Solution

Answer

5(t-2)(t^2+2t+4)

Step 1: Identify the Greatest Common Factor (GCF)

First, look for the greatest common factor (GCF) in the terms of the polynomial. The terms given are 5t^3 and -40. The GCF of 5t^3 and -40 is 5.

Step 2: Factor out the GCF

Next, factor out the GCF from the polynomial. This means we write the polynomial as a product of the GCF and another polynomial: 5(t^3 - 8).

Step 3: Recognize Special Factoring Patterns

Notice that the term inside the parentheses, (t^3 - 8), is a difference of cubes. Recall the formula for factoring the difference of cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2). Here, a = t, and b = 2.

Step 4: Apply the Difference of Cubes Formula

Use the difference of cubes formula to factor:t^3 - 8 = (t - 2)(t^2 + 2t + 4).

Step 5: Combine All Parts

Lastly, combine the factored terms with the GCF factored out previously. The final factored form is: 5 (t - 2) (t^2 + 2t + 4).

Key Concepts

Greatest Common Factor

When you factor polynomials, the first step is often to find the Greatest Common Factor (GCF). This is the largest factor that divides all terms in the polynomial without leaving a remainder.

• With our example, the polynomial is given as \(5t^3 - 40\).
• We need to look at each term and figure out the GCF.
• The terms are \(5t^3\) and -40, and the GCF here is 5.

First, check the numbers in each term. The GCF of 5 and 40 is 5. Next, look at the variables. The term \(t^3\) does not affect the GCF in this context because there is no corresponding term in -40 with \(t\). So, we factor out 5, giving us:5(t^3 - 8)This is a crucial step as it simplifies the polynomial, making it easier to factor further.

Difference of Cubes

After factoring out the GCF, the term inside the parentheses is \(t^3 - 8\). This expression can be factored using the difference of cubes formula.

• Recall the formula for the difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
• In this case, \(t^3 - 8\) fits the pattern where \(a = t\) and \(b = 2\).
• Substituting \(a\) and \(b\) into the formula, we get: \(t^3 - 2^3 = (t - 2)(t^2 + 2t + 4)\).

Understanding this pattern can help you factor other similar expressions quickly. The difference of cubes is a powerful tool in polynomial factorization, so it's helpful to memorize this formula.

Polynomial Factorization

Combining all parts, we conclude the factorization process. After factoring out the GCF and applying the difference of cubes formula, we can piece everything together.

• Initially, we extracted the GCF: \(5(t^3 - 8)\).
• We then factored \(t^3 - 8\) using the difference of cubes: \((t - 2)(t^2 + 2t + 4)\).

The final step involves combining these results:5 (t - 2)(t^2 + 2t + 4)When dealing with polynomial factorization, always check for a GCF first. Then look for special factoring patterns, like the difference of cubes or squares. This systematic approach will make complex problems easier.