Problem 74 - A Graphical Approach to College Algebra

Question

Factor by any method. $$36 a^{2}+60 a+25$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Quadratic Form

The expression to be factored is \(36a^2 + 60a + 25\). This is a quadratic expression in the form \(Ax^2 + Bx + C\), where \(A = 36\), \(B = 60\), and \(C = 25\).

Step 2: Check if it is a Perfect Square Trinomial

We suspect this might be a perfect square trinomial. A perfect square trinomial takes the form \( (ax + b)^2 \), which expands to \(a^2x^2 + 2abx + b^2\). In our case, we have:- \(a^2 = 36\), so \(a = 6\).- \(b^2 = 25\), so \(b = 5\).- We need \(2ab = 60\) to confirm.Calculate \(2 \times 6 \times 5 = 60\), which matches the middle term. Thus, \(36a^2 + 60a + 25\) is indeed a perfect square trinomial.

Step 3: Factor the Perfect Square

Since we verified that \(36a^2 + 60a + 25\) is a perfect square trinomial, we can factor it as:\[(6a + 5)^2\].

Step 4: Verify the Solution

To ensure the factorization is correct, expand \((6a + 5)^2\) to see if it matches the original expression:1. \((6a)^2 = 36a^2\)2. \(2 \times 6a \times 5 = 60a\)3. \(5^2 = 25\)The expansion \(36a^2 + 60a + 25\) confirms that the factorization is correct.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of polynomial that can be expressed as the square of a binomial. This means it takes the form \((ax + b)^2\), where \(a\) and \(b\) are numbers or coefficients. Expanding this binomial, we see that

• the first term is \(a^2x^2\),
• the middle term is \(2abx\), and
• the last term is \(b^2\).

For example, the expression \(36a^2 + 60a + 25\) is a perfect square trinomial. By recognizing the forms \(a^2 = 36\) and \(b^2 = 25\), we identify \(a = 6\) and \(b = 5\). The middle term \(60a\) is verified by calculating \(2 \times 6 \times 5\), which indeed gives \(60a\). Therefore, this expression can be factored into \((6a + 5)^2\). Knowing these elements allows you to spot and factor perfect square trinomials quickly.

Quadratic Expressions

Quadratic expressions are polynomials where the highest degree is 2, meaning they include a squared term. Commonly represented as \(Ax^2 + Bx + C\), quadratic expressions involve three terms:

• \(Ax^2\), the quadratic term,
• \(Bx\), the linear term, and
• \(C\), the constant term.

In our exercise, the expression \(36a^2 + 60a + 25\) fits the structure \(Ax^2 + Bx + C\) with \(A = 36\), \(B = 60\), and \(C = 25\). Recognizing and working with these types of expressions is crucial for factoring and solving quadratic equations. A quadratic expression like this can often be rewritten in a factored form by finding two numbers that multiply to \(A \times C\) and add up to \(B\). However, in this case, recognizing it as a perfect square trinomial offered a more straightforward approach.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into a product of simpler polynomials. For quadratic expressions, this means expressing them as the product of two binomials. Factorization can simplify solving equations, integrating functions, or finding roots. There are several methods to factor quadratics:

• Factoring by grouping, when applicable.
• Completing the square, often used to derive roots.
• Applying the quadratic formula, more of a solution-finding tool than pure factorization.
• Recognizing special forms like perfect square trinomials.

In our problem, the focus was on recognizing a perfect square trinomial, \(36a^2 + 60a + 25\), and factoring it efficiently as \((6a + 5)^2\). Once recognized, this method streamlines problemsolving significantly, avoiding more complex methods. Understanding polynomial factorization in depth is essential, as it provides versatile tools for working with and simplifying polynomial expressions.