Problem 53 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 2 x+4 y-z,-1 $$

Step-by-Step Solution

Answer

Question: Given the product (2x + 4y - z) and one factor -1, find the other factor. Answer: The missing factor is -2x - 4y + z.

Step 1: Identify the given product and factor

The given product is \((2x + 4y - z)\), and the given factor is \(-1\).

Step 2: Use the definition of a factor to set up an equation

If \(-1\) is a factor, we can set up an equation as follows: $$ (-1) \times (MissingFactor) = 2x + 4y - z $$

Step 3: Solve the equation for the missing factor

Divide both sides of the equation by \(-1\): $$ MissingFactor = \frac{2x + 4y - z}{-1} $$

Step 4: Simplify the expression

The expression simplifies to: $$ MissingFactor = -2x - 4y + z $$

Step 5: Check the solution

Multiply the given factor \((-1)\) by the missing factor we found \((-2x - 4y + z)\): $$ (-1) \times (-2x - 4y + z) = 2x + 4y - z $$ The product matches the given product, so our solution is correct: the missing factor is \(-2x - 4y + z\).

Key Concepts

Polynomials

Polynomials are expressions made up of variables, coefficients, and exponents combined using addition, subtraction, and multiplication. They are like building blocks in algebra.
The expression provided in the exercise, \(2x + 4y - z\), is a classic example of a polynomial. Here:

• The variables are \(x\), \(y\), and \(z\).
• The coefficients are the numbers in front of these variables: \(2\), \(4\), and \(-1\) (for \(-z\)).
• The exponents are \(1\) for each of these variables since they are implied when no exponent is written.

Understanding polynomials is vital in algebra because they form the basis for many operations and further algebraic concepts. The degree of this polynomial is 1, indicating it is linear because the highest exponent is 1. Working with polynomials often involves simplifying them, factoring them, or multiplying them.

Factoring

Factoring is an essential process in algebra, which involves breaking down an expression into simpler "factors" that when multiplied together give back the original expression.
In the exercise, the concept of factoring is used to express \(2x + 4y - z\) in terms of a factor setup, \((-1) \times (\text{MissingFactor}) = 2x + 4y - z\).

• The goal is to find the "MissingFactor."
• Factoring here means identifying what can be multiplied with \(-1\) to give the expression \(2x + 4y - z\).

To "unfactor" for the missing factor, you rearrange the equation to solve for the unknown part. This highlights the importance of recognizing patterns, using algebraic rules, and simplifying expressions.

Multiplication in Algebra

Multiplication in algebra involves combining like terms in polynomials and ensuring factors multiply while respecting the algebraic operations. Through this lens, the exercise dives into reverse multiplication – essentially division by a known factor to find the missing piece.

• Mental trick: Think of it as undoing the multiplication by dividing by the known factor to isolate the missing factor.
• The known factor is \(-1\). So, to find what paired with \(-1\) to make \(2x + 4y - z\), you divide the polynomial by \(-1\).
• This is shown as: \(\text{MissingFactor} = \frac{2x + 4y - z}{-1}\) simplifies to \(-2x - 4y + z\).

Multiplication might be about combining expressions, but sometimes it means unpacking them by understanding which parts come together to form the whole. This skill is crucial for higher mathematical functions and equations, where multiplication and division interact seamlessly.