Problem 26 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 4 y^{2}-10 y-12, \quad 2 $$

Step-by-Step Solution

Answer

Question: Given the product $$4y^2 - 10y - 12$$ and a known factor of $$2$$, find the other factor. Answer: The other factor is $$2y^2 - 5y - 6$$.

Step 1: Write the given product and factor

Given: Product: $$4y^2 - 10y - 12$$ Known factor: $$2$$

Step 2: Divide product by known factor

We want to find the other factor, so we will divide the given product by the known factor: $$ \frac{4y^2 - 10y - 12}{2} $$

Step 3: Simplify the expression

Now we simplify the expression as much as possible: $$ \begin{aligned} \frac{4y^2 - 10y - 12}{2} &= \frac{4}{2}y^2 - \frac{10}{2}y - \frac{12}{2} \\ &= 2y^2 - 5y - 6 \end{aligned} $$

Step 4: The other factor

The other factor is the result of the division: $$ 2y^2 - 5y - 6 $$ Hence, the other factor is $$2y^2 - 5y - 6$$.

Key Concepts

Polynomial Division

Polynomial division is akin to the division you've learned with numbers, yet it's a tool for dividing one polynomial by another, leading to a simpler form or breaking it down into factors. Similar to numeric long division, we divide the highest degree term of the numerator by the highest degree term of the denominator and repeat this process for each term.

In our exercise, we divide the polynomial function, which is a quadratic expression consisting of terms with powers of two, one, and a constant, by a known factor. The essence of this process is simplifying the complex polynomial into a more manageable form, shedding light on its constituent parts. Think of it as unpacking a suitcase to see what's inside, ensuring you don't miss anything important. This step is crucial as it paves the way to finding the other factor of the polynomial.

Factorization

Factorization is the process of breaking down an algebraic expression into the simplest building blocks—factors—that, when multiplied together, give us the original expression. It's comparable to finding out the recipe for a cake by deconstructing it into its ingredients.

For the given exercise, we're essentially reverse-engineering the polynomial, splitting it into a product of its factors. One factor has been given, which leads us to discover the other through polynomial division. To continue the analogy, if you know that eggs are part of the recipe (the known factor), and you have the whole cake (the product), factorization helps you figure out what other ingredients went into the making of that cake.

Simplifying Expressions

Simplifying expressions involves reducing an algebraic expression to its simplest form, making it easier to work with or understand. This process could mean anything from combining like terms to factoring expressions or using algebraic identities to simplify.

Think of simplifying as tidying up your workspace; it's all about getting rid of unnecessary clutter and organizing what's left in a straightforward, accessible manner. In our example, simplification comes after dividing the polynomial by the known factor. Each term of the polynomial is divided by two, stripping away the layers of complexity, and leaving us with a clearer picture of the algebraic expression we are working with.

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra, representing numbers and their relationships through variables and constants. They are the phrases and clauses of the mathematical language, telling us about the quantities involved and their interconnections.

In our exercise, the algebraic expression is the polynomial we're investigating, consisting of terms with variables raised to different powers and combined using arithmetic operations. Understanding these expressions requires a deep appreciation of how algebraic operations work, setting the stage for tasks such as simplifying, factoring, and solving equations. Comprehending algebraic expressions and mastering their manipulation is fundamental to progressing in your study of algebra.