Problem 108 - Introductory Algebra for College Students

Question

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$4 a y^{3}-12 a y^{2}+9 a y$$

Step-by-Step Solution

Answer

The fully factored polynomial is \(a y (2y - 3)^2\).

Step 1: Identify Common Factors

Examine the polynomial and observe that 'a y' is a common factor throughout each term. We have: \(4 a y^{3}-12 a y^{2}+9 a y = a y (4y^2 - 12y + 9)\)

Step 2: Factorize Further

The quadratic function inside the brackets can be factorized further as \((2y - 3)^2\). Therefore, the completely factored form of the polynomial is: \(a y (2y - 3)^2\).

Step 3: Check the Solution

In order to verify that our factoring is correct, we can multiply the factors back together to obtain the original expression. Multiplying \(a y (2y - 3)^2\) or \(a y [(2y - 3)(2y - 3)]\), we indeed obtain the original expression, \(4 a y^3 - 12 a y^2 + 9 a y\).

Key Concepts

Polynomial Factoring

Polynomial factoring involves rewriting a polynomial as a product of simpler polynomials. This process helps in simplifying expressions and solving polynomial equations. Factoring is a fundamental concept in algebra that can reveal zeros or roots of polynomials. In the given exercise, the polynomial is broken down into common factors and simpler quadratic expressions.

• Start by identifying any common factors in all terms of the polynomial.
• Next, if the polynomial inside the brackets can be factored further, continue with this process.
• The goal is to express the polynomial as a product of its factors.

Factoring is especially useful in solving equations because setting each factor equal to zero can help find the values of variables that satisfy the equation. As seen in the exercise, factored form makes it easier to check solutions by multiplying back to obtain the original expression.

Quadratic Function

A quadratic function is a polynomial of degree 2, represented in the form \[ax^2 + bx + c\]. Quadratics are important in many real-world applications like physics and statistics. In the exercise, the quadratic expression found inside the brackets is \[4y^2 - 12y + 9\]. The method for factoring quadratics such as this involves finding two numbers that multiply to give 'ac' (4 times 9 in this example) and add to give 'b' (12 here).

• This particular quadratic factors into \[(2y - 3)^2\], meaning that the same factor is repeated twice.

Quadratic functions typically graph as parabolas. Factoring these functions not only helps solve equations but also aids in understanding their visual graphs. Recognizing a quadratic within a polynomial, as in our exercise, eases the problem-solving process greatly.

Common Factors

Common factors are terms or expressions that are present in each term of a polynomial. Identifying these factors is the initial key step in the factoring process. A common mistake is ignoring this step, which can complicate further factoring.

• In the polynomial from the exercise, \(4ay^3 - 12ay^2 + 9ay\), the common factor is \(ay\).
• Factoring this out simplifies the polynomial to \(ay(4y^2 - 12y + 9)\).

Finding common factors is about looking for the greatest common divisor in terms or analyzing each coefficient. It's the first simplification that often leads to correctly breaking down a polynomial into manageable pieces, making solving the polynomial more efficient and understandable. This concept is critical in algebra and makes more complex operations in calculus feasible by reducing polynomial expressions initially.