Problem 4 - Elementary Algebra

Question

Determine the missing factor. $$ -21 x 2 y+7 x y 2-49 x y=-7 x y(\quad ? \quad) $$

Step-by-Step Solution

Answer

The missing factor is \(2y + 7\).

Step 1: Identify the Expression Structure

The expression on the left-hand side of the equation is \(-21xy^2 + 7xy^2 - 49xy\). Notice that each term has a common factor of \(-7xy\). This is evident because the terms can be rewritten as factors of \(-7xy\) by dividing them individually by \(-7xy\).

Step 2: Factor Each Term

Let's factor each term one by one:- \(-21xy^2 \div -7xy = 3y\), because \(-21 / -7 = 3\) and \(y^2 / y = y\).- \(7xy^2 \div -7xy = -y\), because \(7 / -7 = -1\) and \(y^2 / y = y\).- \(-49xy \div -7xy = 7\), because \(-49 / -7 = 7\) and \(xy/xy = 1\).These give us the transformed and factored components.

Step 3: Write the Overall Factorization

Now, using these factors, the expression can be rewritten within the parentheses multiplied by \(-7xy\): \(-7xy(3y - y + 7)\).

Step 4: Simplify the Parentheses

Simplify the expression inside the parentheses: \[3y - y + 7 = 2y + 7\].

Step 5: Finalize the Solution

Thus, the original equation becomes:\(-21xy^2 + 7xy^2 - 49xy = -7xy(2y + 7)\).The expression inside the parenthesis \((?\)) is \(2y + 7\).

Key Concepts

Common Factors

Identifying common factors is a fundamental concept in algebra that simplifies expressions by dividing each term by a shared element. In the given expression \[-21xy^2 + 7xy^2 - 49xy,\] every term can share a common factor of \(-7xy\).

Recognizing common factors helps in pulling out a part of the expression that is shared among all terms, making it easier to handle:

• Supports the simplification process.
• Aids in the factorization of polynomial expressions.
• Makes solving equations more manageable.

By factoring out \(-7xy\)from each term, you reduce the complexity of the expression, leading to a form that is simpler to manipulate and solve.

Polynomial Expressions

Polynomial expressions consist of variables and coefficients, connected through operations of addition, subtraction, and multiplication. They often appear in mathematical equations and can be subjected to operations like addition and subtraction. Our expression \(-21xy^2 + 7xy^2 - 49xy\)serves as a clear example of a polynomial.

Understanding polynomial expressions involves recognizing:

• The degree of the polynomial, determined by the highest power of the variable.
• Individual terms, each characterized by a coefficient, a variable part, and an exponent.
• Operations that can be performed, such as combining like terms or factorizations.

Working with polynomial expressions also means being able to manipulate them using algebraic rules, to simplify and solve them efficiently. Such manipulation includes identifying patterns like common factors or using properties like the distributive property for expansion or factorization.

Algebraic Factorization

Algebraic factorization is the process of expressing a polynomial as a product of simpler polynomials or factors. By breaking down complex expressions into manageable parts, it enables easier solving and simplification. Our task involved expressing the original polynomial expression using its common factor: \(-7xy(2y + 7)\).

Key points about algebraic factorization include:

• Finding the greatest common factor (GCF) to simplify expressions efficiently.
• Rewriting expressions to their simplest factor form allows for more straightforward computation and understanding.
• It is crucial in solving quadratic equations and expressions, making it easier to find roots or simplify further.

Factorization not only simplifies expressions, but also paves the way for solving more intricate algebraic problems, contributing significantly to understanding algebra at deeper levels.