Problem 48 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 8 x^{2}+12 x,-4 x $$

Step-by-Step Solution

Answer

Answer: The other factor is -2x - 3.

Step 1: Write the Product and Factor

We are given the product: $$ Product = 8x^2 + 12x $$ And the given factor: $$ Factor = -4x $$

Step 2: Set Up the Polynomial Division

Now we will set up the polynomial division with the product as the dividend and the given factor as the divisor: $$ \frac{8x^2 + 12x}{-4x} $$

Step 3: Divide the Polynomial

Divide the terms of the product by the divisor (-4x): $$ \frac{8x^2}{-4x} + \frac{12x}{-4x} $$

Step 4: Simplify Each Fraction

Simplify each fraction: $$ (-2x) + (-3) $$ So, the other factor is -2x -3.

Step 5: Write the Final Answer

The other factor is: $$ -2x - 3 $$

Key Concepts

Polynomial Algebra

Polynomial algebra involves operations with polynomials, including addition, subtraction, multiplication, division, and factoring. In the context of the exercise provided, we're specifically looking at polynomial division, which is the process of dividing a polynomial (dividend) by another polynomial (divisor) to find a quotient and possibly a remainder.

When we are given a product such as \( 8x^2 + 12x \) and a factor like \( -4x \), we use polynomial division to find the other factor. This other factor, when multiplied by \( -4x \), would give us the original polynomial product. In this scenario, polynomial division simplifies to dividing each term of the polynomial by \( -4x \), leading us to discover the other factor.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into a product of simpler polynomials or factors. It's a key concept in algebra that can simplify many types of problems such as solving equations or simplifying expressions.

For instance, in our example \( 8x^2 + 12x \), finding the other factor given \( -4x \) involves the reverse process of factoring; we're essentially determining what must be multiplied by \( -4x \) to get the original polynomial. Recognizing common terms, utilizing the distributive property, and applying factoring techniques (such as pulling out the greatest common factor or GCF) are fundamental in solving these types of problems. The simplicity of the solution can often be enhanced when understanding these factoring techniques.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a part of algebra that focuses on making expressions easier to understand and work with by reducing them into their simplest form. This often involves combining like terms, reducing fractions, and factoring.

In the division step of our exercise, each term of \( 8x^2 + 12x \) was divided by the factor \( -4x \). This is a prime example of simplifying an algebraic expression, as we're reducing a more complex expression into its simplest terms. The terms \( 8x^2 \/ -4x \) and \( 12x \/ -4x \) both simplify by dividing coefficients and canceling out common factors, leading to the quotient \( -2x - 3 \). Mastery of simplification is vital to algebra, as it not only makes problems easier to tackle, but also helps in understanding the underlying structure of the expressions.