Problem 30 - Elementary Algebra

Question

For the following problems, factor the polynomials. $$ a x^{2}+a x+a $$

Step-by-Step Solution

Answer

Answer: The factored form of the polynomial is \(a (x^{2} + x + 1)\).

Step 1: Identify common factors

For the provided polynomial \(ax^{2} + ax + a\), we can see that the variable \(a\) is a common factor in all three terms.

Step 2: Factor out the common factor

Next, we want to factor out the common factor \(a\) from the polynomial. This is done by dividing each term by \(a\): $$ a x^{2}+ax+a = a \cdot (x^{2} + x + 1) $$

Step 3: Check the result

Finally, check the factored polynomial by multiplying it back together: $$ a (x^{2} + x + 1) = ax^{2} + ax + a $$ The expanded form of the factored polynomial matches the original polynomial, so the factoring is correct. The factored polynomial is: $$ ax^{2} + ax + a = a (x^{2} + x + 1) $$

Key Concepts

Common Factor

When we talk about factoring polynomials, the concept of a 'common factor' is essential. This refers to a term that is shared by each component of a polynomial. Identifying the common factor is your first step towards simplifying the expression.

For instance, if we look at the polynomial given in our exercise, each term includes the variable 'a'. This repetition shows that 'a' is a common factor. By extracting this common factor from each term, we simplify the algebraic expression into a more condensed form. It's like breaking down a bigger problem into smaller, more manageable pieces, ultimately making the overall problem easier to understand and solve.

This strategy does not only apply to a single variable; sometimes you'll find a numeric coefficient or even a longer expression that can serve as a common factor. It's all about looking for patterns and similarities across terms in the polynomial.

Algebraic Expressions

An algebraic expression is a mathematical phrase that combines numbers, variables, and arithmetic operations. Polynomials, such as the one in our example, are a type of algebraic expression that includes terms added together. Each term can be a variable, a number, or a combination of both, often multiplied together.

The beauty of algebraic expressions lies in their versatility; they are the building blocks for modeling real-world situations, such as the path of a falling object or the growth of an investment. By mastering the art of manipulation and simplification of these expressions through techniques like factoring, you can solve intricate problems and unlock patterns that reveal deeper mathematical insights.

It's important to understand that the expression's value can change depending on the values assigned to its variables. Hence, algebraic expressions are like formulas waiting to be applied to various situations, depending on what we know or want to find out.

Polynomial Division

Dividing a polynomial by its common factor is a process that mirrors long division with numbers. When we factor, what we're doing is essentially dividing the polynomial by a common factor to simplify it into a product of factors.

How Does it Work?

Just as we did in our example exercise, we divide each term of the polynomial by the common factor, which in this case is 'a'. The result is a simpler expression, where 'a' is then coupled as a multiplier to the simplified polynomial.

• Divide each term individually.
• Write the dividend (what you're dividing) as a product of the divisor (the common factor) and the quotient (simplified expression).
• Verify by multiplying the factors back together.

The ability to divide polynomials not only aids in factoring but also prepares students for more complex algebraic problems, such as synthetic division and solving polynomial equations. As with long division, practice makes perfect, and getting comfortable with recognizing common factors simplifies seemingly complex problems tremendously.