Problem 90 - College Algebra with Modeling and Visualization

Question

Factor the expression. \(16 a^{3}+8 a^{2} b+a b^{2}\)

Step-by-Step Solution

Answer

The expression factors to \(a(4a + b)^2\).

Step 1: Identify the Greatest Common Factor (GCF)

The first step in factoring any expression is to look for the greatest common factor in all terms. For the expression \(16a^3 + 8a^2b + ab^2\), each term shares a common factor of \(a\). Factor out \(a\) from each term.

Step 2: Factor Out the GCF

After identifying the GCF as \(a\), divide each term by \(a\) and factor it out, resulting in:\[a(16a^2 + 8ab + b^2)\]This simplifies the expression and prepares it for further factoring.

Step 3: Factor the Quadratic Expression

Now, focus on the remaining quadratic expression \(16a^2 + 8ab + b^2\). To factor this, we recognize that it is a perfect square trinomial since the first and last terms are perfect squares and the middle term is twice the product of their roots:\[16a^2 = (4a)^2, \, b^2 = (b)^2, \, \text{and} \, 8ab = 2 \cdot 4a \cdot b\] So, it can be factored as \[(4a + b)^2\]

Step 4: Write the Complete Factored Form

Combine the factored terms from Steps 2 and 3. The original expression \(16a^3 + 8a^2b + ab^2\) is therefore factored as:\[a(4a + b)^2\]This is the completely factored form of the expression.

Key Concepts

Greatest Common Factor (GCF)

Finding the Greatest Common Factor (GCF) is a crucial step in simplifying expressions effectively. The GCF is the highest factor that divides all terms of a given polynomial without any remainder. It's like finding the biggest ingredient that each term in the expression shares, allowing us to "factor it out" to simplify initially complex expressions.

In the example exercise, the terms are:

• \(16a^3\)
• \(8a^2b\)
• \(ab^2\)

Each term here contains the factor \(a\). Thus, \(a\) is the GCF for this polynomial.

To simplify, divide each term by \(a\), which results in factoring it out of the expression. This consolidation is the first step toward making factoring more manageable, setting the stage for more targeted operations on the remaining polynomial.

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that factors into a squared binomial. It's like the perfect blend of terms that form a tidy square when multiplied. Recognizing these trinomial forms helps streamline the factoring process.

A trinomial is a perfect square if:

• The first term is a perfect square, meaning it is the square of another number or expression.
• The last term is also a perfect square.
• The middle term is twice the product of the roots of these squares.

In our exercise:

• \(16a^2\) is \((4a)^2\)
• \(b^2\) is \((b)^2\)
• \(8ab = 2 \times 4a \times b\)

Hence, the expression \(16a^2 + 8ab + b^2\) becomes \((4a + b)^2\). Factoring it as a perfect square trinomial allows ease of solution and reflects the structure back to its core binomial form.

Quadratic Expression

Quadratic expressions are polynomials of degree two, typically in the form \(ax^2 + bx + c\). They form the backbone of algebraic operations and often crop up in various forms across mathematics.

The challenge lies in recognizing different ways these expressions factor into simpler, more manageable parts. In the instance of our exercise, the quadratic expression \(16a^2 + 8ab + b^2\) doesn't appear in a typical \(ax^2 + bx + c\) form at first glance. Instead, as factored into \(a(4a + b)^2\), it exemplifies a perfect square trinomial, showcasing how quadratics can be expressed as the square of a binomial.

Understanding these conversion dynamics is critical for efficiently solving equations and identifying patterns in algebraic structures. Always look for common patterns within the expression, like perfect squares or similar terms, to simplify and solve.