Problem 9 - Elementary Algebra

Question

For the following problems, use the grouping method to factor the polynomials. Some polynomials may not. be factorable using the grouping method. $$ x y+x+3 y+3 $$

Step-by-Step Solution

Answer

Answer: The factored form of the polynomial is \((x + 3)(y + 1)\).

Step 1: 1: Rearrange terms

First, let's rearrange the terms in the polynomial: $$ x y+x+3 y+3 $$ Rearrange to group the terms with common factors: $$ x y + 3y + x + 3 $$

Step 2: 2: Factor out common factors from each group

Now, let's factor out the common factors from each group: In the first group \((x y + 3y)\), the common factor is \(y\). So, we factor out \(y\): $$ y(x + 3) $$ In the second group \((x + 3)\), there is no common factor to be taken out, so we keep it as is: $$ (x + 3) $$

Step 3: 3: Check for common factors in the new groups

Now, let's look at the two groups \((y(x+3))\) and \((x+3)\). Notice that they both have a common factor \((x+3)\).

Step 4: 4: Factor the common factor

Since \((x+3)\) is a common factor in both of our groups, we can factor it out: $$ (x + 3)(y+1) $$ The factored form of the given polynomial using the grouping method is: $$ (x + 3)(y+1) $$

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical expressions consisting of variables and coefficients. The variables within a polynomial expression are combined using addition, subtraction, and multiplication. Each distinct term in these expressions is formed by multiplying a constant number by the variable term raised to a power.To break it down:

• The term "polynomial" derives from the Greek words "poly" (many) and "nomos" (parts or terms).
• A simple example of a polynomial is \(3x^2 + 2x + 1\).
• Polynomials can have one or more terms, with each term consisting of a coefficient (like 3 or 2) and a variable (like \(x\)).
• We typically identify the highest power of the variable present in the expression to denote the degree of the polynomial, such as degree 2 in the example \(3x^2\).

Polynomials are foundational in algebra and have countless applications in science and engineering.

Algebraic Factoring

Factoring in algebra is the process of rewriting polynomials as a product of their simpler components. It is similar to decomposing a number into its prime factors, but instead, we work with algebraic expressions.In algebraic factoring, you are trying to find two or more expressions that multiply together to bring you back to the original polynomial.

• By identifying and factoring out common terms, we simplify the polynomial and make it easier to work with.
• For example, the polynomial \(x^2 + 5x + 6\) factors to \((x + 2)(x + 3)\).
• Polynomials can have simple factors or more complex ones, involving multiple steps or attempting other methods if the specific approach like 'grouping' isn't feasible.

Factoring is crucial for solving polynomial equations, simplifying expressions, and finding roots.

Grouping Method

The grouping method is a factoring technique particularly useful when dealing with polynomials having four or more terms. It involves arranging and grouping terms in a way that common factors can be easily identified.Here's how the method works:

• Examine the polynomial: \(xy + x + 3y + 3 \).
• Rearrange the terms: Group them as \((xy + 3y) + (x + 3) \).
• Factor the groups: Notice \(y(x + 3)\) in the first and keep \((x + 3)\) in the second as no common factor exists for \(x\) and \(3\).
• Identify a common binomial factor: Both groups have \((x + 3)\), factor it out to get \((x + 3)(y + 1)\).

The grouping method simplifies polynomials to a form that is easier to understand or solve. It's a handy tool in algebra, enabling us to uncover factors quickly even in polynomials with multiple terms.