Problem 39 - Elementary Algebra

Question

For the following problems, list, if any should appear, the common factors in the expressions. $$ 45 a b^{2}+9 b^{2} $$

Step-by-Step Solution

Answer

Answer: The factored form of the expression is \(9b^2(5a + 1)\).

Step 1: Identifying the common factors in both terms

Firstly, look at the coefficients of the given expression: \(45\) and \(9\). We can find the Greatest Common Divisor (GCD) between them. The GCD of \(45\) and \(9\) is \(9\). Now, let's check the variables in both terms. We have \(a\) and \(b^2\). The common variable is \(b^2\), as it appears in both terms.

Step 2: Factoring out the common factors

Now that the common factors are identified, we can factor them out of the expression. The common factors are \(9\) and \(b^2\). By factoring them out, we get: $$ 9b^2(5a + 1) $$ The factored expression is: \(9b^2(5a + 1)\).

Key Concepts

Greatest Common Divisor

When factoring algebraic expressions, the concept of the Greatest Common Divisor (GCD) is crucial. The GCD of two or more numbers is the largest number that divides all of them without leaving a remainder. Imagine it as the 'biggest piece of the pie' that can be evenly shared among them. In the solution to our exercise, the GCD is found between the coefficients 45 and 9, which is 9.

Understanding the GCD is not just about finding a number, though; it's about recognizing patterns and simplifying expressions. To determine the GCD of two numbers, you could list out the factors of each number and find the highest one they have in common, or you could use the Euclidean algorithm – a systematic method to compute the GCD. Using the GCD allows you to 'pull out' what's shared between terms and make the expression tidier, just like condensing a fraction to its simplest form. It's an essential step in factoring expressions and one that can simplify many aspects of algebra.

Algebraic Factors

Algebraic factors are the building blocks of algebraic expressions, much like how lego pieces come together to build complex structures. They are the parts of a term that, when multiplied together, give the term itself. For instance, in our exercise, the term '45ab^2' can be viewed as the product of the factors '45', 'a', and 'b^2'.

Identifying Common Algebraic Factors

Identifying common algebraic factors in terms like '45ab^2' and '9b^2' is like finding the common ingredients in different recipes. If both recipes call for eggs, that's your common factor. Similarly, noticing that both terms have 'b^2' gives you a common algebraic factor to extract. Once you've identified these commonalities, factoring them out simplifies the expression, leading to a more comprehensible and elegant algebraic statement.

Coefficient

In algebra, the coefficient is the numerical factor of a term that contains a variable. You can think of it as the number that stands before the variable, giving us information about its 'weight' or influence in an expression. In the exercise's term '45ab^2', '45' is the coefficient, setting the stage for how strongly 'a' and 'b^2' play a role.

But coefficients aren't just numbers; they are the anchors that help us navigate the algebraic sea. When factoring, coefficients provide critical clues for finding common factors. They can be manipulated, factored out, or even become variables in more complex algebra. By identifying the GCD of two coefficients, as we did with '45' and '9', we can factor algebraic expressions more effectively and streamline our work into a form that's easier to understand and apply – simplifying both solutions and our mathematical journey.