Problem 61 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 16 a^{5}-54 a^{2} $$

Step-by-Step Solution

Answer

The expression factors as \(2a^2(2a - 3)(4a^2 + 6a + 9)\).

Step 1: Identify Common Factors

First, look for the greatest common factor (GCF) in the expression \(16a^5 - 54a^2\). Both terms have \(a^2\) as a common factor.

Step 2: Factor Out the GCF

Factor \(a^2\) out of each term in the expression: \[ 16a^5 - 54a^2 = a^2(16a^3 - 54) \]

Step 3: Simplify Inside the Parenthesis

The expression \(16a^3 - 54\) can be factored further. Notice that both coefficients 16 and 54 are divisible by 2.

Step 4: Factor Out the Numerical GCF Inside Parenthesis

Factor out the numerical GCF (2) from \(16a^3 - 54\):\[ a^2(16a^3 - 54) = a^2 imes 2(8a^3 - 27) \] which simplifies to \[ 2a^2(8a^3 - 27) \]

Step 5: Recognize as Difference of Cubes

Recognize that \(8a^3 - 27\) is a difference of cubes: \((2a)^3 - 3^3\). The difference of cubes formula is: \(a^3 - b^3 = (a-b)(a^2+ab+b^2)\).

Step 6: Apply the Difference of Cubes Formula

Apply the difference of cubes formula: \[ 8a^3 - 27 = (2a - 3)((2a)^2 + 2a(3) + 3^2) \] Simplify the terms: \[ (2a - 3)(4a^2 + 6a + 9) \]

Step 7: Write the Final Factored Expression

Combine all the factors identified: The expression \(16a^5 - 54a^2\) factors to its simplest form as: \[ 2a^2(2a - 3)(4a^2 + 6a + 9) \]

Key Concepts

Greatest Common Factor

In algebra, finding the greatest common factor (GCF) is a vital step in simplifying expressions or polynomials. The GCF is the largest expression that evenly divides each term of the polynomial. It's a way to "pull out" a common element which simplifies the original expression.
For example, take the expression given:

• Terms: \(16a^5\) and \(54a^2\)
• First, focus on the coefficients. The numbers are 16 and 54. The greatest common divisor (GCD) of these two numbers is 2.
• Second, check the variable parts. Both terms have \(a^2\) as a common factor.
• Combine these observations, making the GCF for \(16a^5 - 54a^2\) be \(2a^2\).

By factoring out the GCF \(2a^2\), you're left with a simpler expression inside the parenthesis, which can be further analyzed or factored down in subsequent steps.

Difference of Cubes

Recognizing special patterns like the difference of cubes can significantly simplify polynomial factorization. The difference of cubes formula is\(a^3 - b^3 = (a-b)(a^2+ab+b^2)\). Spotting this can be crucial in simplifying expressions.
Consider our current factored expression:

• Inside the parenthesis, after factoring out the GCF, is \(8a^3 - 27\).
• This is a difference of cubes because \(8a^3\) is \((2a)^3\) and \(27\) is \(3^3\).
• Apply the difference of cubes formula to \((2a)^3 - 3^3\).
• The result is the factorization: \((2a - 3)((2a)^2 + 2a(3) + 3^2)\).

Through application and simplification, this powerful pattern allows us to further break down polynomials into simpler, more manageable factors.

Polynomial Factorization

Solving polynomial equations often involves factoring them into simpler pieces, which represent the polynomial's behavior more readily. Factorization is both an art and skill in algebra.
In the exercise provided:

• The full expression \(16a^5 - 54a^2\) initially seems complex, but breaking it down step by step helps find its simplest form.
• The greatest common factor, \(2a^2\), is factored out first, effectively reducing complexity.
• Next, recognizing and applying the difference of cubes on \(8a^3 - 27\) allows further breakdown, resulting in\((2a - 3)(4a^2 + 6a + 9)\).

The whole factored expression is \(2a^2(2a - 3)(4a^2 + 6a + 9)\).
These steps illustrate how spotting patterns, using logical steps, and understanding polynomial families enable tackling even seemingly tough problems with confidence and accuracy.