Problem 71 - Beginning and Intermediate Algebra

Question

Factor completely. $$9 y^{2}(z-10)^{3}+76 y(z-10)^{3}+32(z-10)^{3}$$

Step-by-Step Solution

Answer

The completely factored form of the given expression is \((z-10)^3(9y^2 + 76y + 32)\).

Step 1: Identify common factor

In this expression, we see that each term has the common factor \((z-10)^3\). We will factor this out.

Step 2: Factor out the common factor

Now, factor out the common factor \((z-10)^3\): \((z-10)^3(9y^2 + 76y + 32)\)

Step 3: Simplify the remaining expression in parentheses (if possible)

Observe the expression within the parentheses: \(9y^2 + 76y + 32\) It is a quadratic in terms of \(y\). We will check if it can be factored further by finding two numbers whose product is equal to the product of the first term coefficient (9) and the last term (32), and whose sum is equal to the middle term coefficient (76). In this case, such numbers do not exist, meaning the quadratic cannot be factored further.

Step 4: Write the final answer

Our factored expression is: \((z-10)^3(9y^2 + 76y + 32)\) This is the completely factored form of the given expression.

Key Concepts

Common Factor

Finding a common factor is an essential skill when simplifying expressions. In the given polynomial, each term includes the part

• \((z-10)^3\)

. This means

• \((z-10)^3\)

is a common factor across all terms. Factoring this out from each term simplifies the equation. By factoring out the common elements, the expression becomes easier to handle and further factor if needed. This approach reduces complexity, making it clearer what remains and if anything else can be further broken down.

Quadratic Expression

A quadratic expression is characterized by its highest degree of 2, often represented as

• \(ax^2 + bx + c\)

. In this case, after factoring out the common factor \((z-10)^3\), we are left with the quadratic expression

• \(9y^2 + 76y + 32\).

Quadratics can potentially be factored further if two numbers can be found that multiply to provide the product of the first and last coefficients and add up to the middle coefficient. Understanding how to identify and handle quadratic expressions is crucial in polynomial factoring.

Factoring Techniques

When it comes to factoring, several techniques can be applied. First, identifying any common factors, like in this example, where we factored out

• \((z-10)^3\)

, is fundamental. For quadratic expressions, different methods such as:

• trial and error,
• the use of the quadratic formula,
• or completing the square

may be used if straightforward factoring is not possible. Here, the quadratic was checked for further factoring but was found to be non-factorable using simple integer methods due to the nature of its coefficients.

Completely Factored Form

Reaching the completely factored form signifies that the expression cannot be reduced any further. In the problem, after factoring out the common factor

• \((z-10)^3\)

, and checking the remaining quadratic

• \(9y^2 + 76y + 32\)

, we confirm that no integers satisfy the criteria for further factoring. Therefore, the expression

• \((z-10)^3(9y^2 + 76y + 32)\)

is in its most simplified or completely factored form. Identifying this form is key to solving or simplifying polynomials entirely.