Problem 88 - College Algebra with Modeling and Visualization

Question

Factor the expression. \(25 a^{2}+60 a b+36 b^{2}\)

Step-by-Step Solution

Answer

The expression factors to \((5a + 6b)^2\).

Step 1: Identify the Form

The expression is in the form of a quadratic trinomial: \(25a^2 + 60ab + 36b^2\). Notice that it fits the pattern \(Ax^2 + Bxy + Cy^2\).

Step 2: Check if it's a Perfect Square Trinomial

To factor the trinomial, check if it is a perfect square. The expression will be a perfect square if \(A\), \(B\), and \(C\) satisfy the equation: \((rac{B}{2})^2 = AC\). In this case, \((rac{60}{2})^2 = 25 \times 36\); both equal 900, confirming it is a perfect square.

Step 3: Write as a Perfect Square

Since the trinomial is a perfect square, express it in the form \((D x + Ey)^2\). We'll find \(D\) and \(E\) by taking the square root of \(A\) and \(C\), respectively: \(D = \sqrt{25} = 5\) and \(E = \sqrt{36} = 6\). Thus, the trinomial can be expressed as: \((5a + 6b)^2\).

Step 4: Verify the Factorization

Expand \((5a + 6b)^2\) to verify: \((5a + 6b)(5a + 6b) = 25a^2 + 30ab + 30ab + 36b^2 = 25a^2 + 60ab + 36b^2\). The expression matches the original, confirming the factorization is correct.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special kind of quadratic trinomial. It takes on a specific form that allows it to be factored into a squared binomial. Understanding this concept is key to simplifying and solving equations effectively.

In a perfect square trinomial, we have expressions like \[Ax^2 + Bxy + Cy^2\]which can be rewritten as:\[(Dx + Ey)^2\].
Here's how it works:

• Identify if the trinomial can be expressed as a square. This involves checking if \[(\frac{B}{2})^2 = AC\],where \(A\), \(B\), and \(C\) are coefficients of the trinomial.


• If this equation holds true, then the trinomial is a perfect square. This was demonstrated in our original problem, where both sides equaled 900.


• Once identified, finding the square roots of \(A\) and \(C\) will help us find \(D\) and \(E\) needed to write the expression as a perfect square.

Recognizing and rewriting a perfect square trinomial simplifies expressions and aids in solving equations efficiently.

Quadratic Trinomial

A quadratic trinomial is a polynomial consisting of three terms with the highest degree being two. It is generally expressed in the form \[Ax^2 + Bx + C\], where \(A\), \(B\), and \(C\) are constants.

Quadratic trinomials are a broad category, encompassing many polynomial expressions you encounter in algebra.
Key points include:

• The main term, \(Ax^2\), signifies that the polynomial is quadratic.


• The linear term, \(Bx\), introduces a single power of \(x\) apart from the quadratic term.


• The constant term, \(C\), is free of any variable.

These polynomials form the foundation of factorization and are vital for understanding various algebraic concepts. Techniques like completing the square or the quadratic formula often involve working with these trinomials. Recognizing them helps in choosing the appropriate method for solving or simplifying equations.

Polynomial Factorization

Polynomial factorization is a process used in algebra to reduce polynomials into a product of simpler factors. This makes solving polynomial equations more manageable and is a core component of algebra.

Factorization involves these steps:

• Identify the structure: Determine the type of polynomial you are working with. For the quadratic trinomial, it will help to see if it is a perfect square, like in our exercise.


• Apply methods: Use methods such as the greatest common factor (GCF), difference of squares, or for specific forms like perfect squares, recognizing and rewriting them appropriately.


• Verify results: After factorizing, multiply the factors to ensure they reconstruct the original polynomial. This ensures accuracy in your factorization.

Understanding polynomial factorization helps to simplify expressions, solve polynomial equations, and tackle algebraic problems with confidence. It's essential in advanced topics too, as seen with factoring perfect square trinomials.