Problem 7 - A Graphical Approach to College Algebra

Question

Factor the greatest common factor from each polynomial. $$(2 y-3)(y+2)+(y+5)(y+2)$$

Step-by-Step Solution

Answer

The polynomial factors to \((y+2)(3y+2)\).

Step 1: Identify Common Factors

First, observe that both terms, \((2y-3)(y+2)\) and \((y+5)(y+2)\), contain the common binomial factor \((y+2)\). This common factor will be factored out in the next step.

Step 2: Factor Out the Common Factor

Factor \((y+2)\) out from the expression. This can be done by writing the expression as a product of the common factor \((y+2)\) and the sum of the remaining terms. The expression becomes:\[(y+2)((2y-3) + (y+5))\]

Step 3: Simplify Inside the Parentheses

Add the terms inside the parentheses. Combine \((2y-3)\) and \((y+5)\):\[(2y-3) + (y+5) = 2y + y - 3 + 5 = 3y + 2\]

Step 4: Write the Final Factored Form

Now that the inner expression has been simplified, the entire polynomial can be written as:\[(y+2)(3y+2)\]Thus, the final factored form of the polynomial is \((y+2)(3y+2)\).

Key Concepts

Greatest Common Factor

In mathematics, when we talk about the Greatest Common Factor (GCF), we refer to the largest factor that divides two or more numbers or terms without leaving a remainder. In the context of polynomials, like in the exercise provided, the GCF is the largest polynomial that is a factor of two or more polynomials.Finding the GCF in polynomials involves identifying common expressions or terms shared between the different parts of a polynomial expression. In the exercise provided, the focus is on the binomial factor \((y+2)\).

• The first term—\((2y-3)(y+2)\)—contains the factor \((y+2)\).
• The second term—\((y+5)(y+2)\)—also contains \((y+2)\) as a factor.

By factoring \((y+2)\) out of the entire expression, it becomes the largest factor (i.e., the GCF) common to both polynomial terms, thereby simplifying the expression.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients. They are formed by adding or subtracting terms, each of which is a product of a constant and one or more variables raised to a whole number power.In the exercise, the expression consists of two polynomial terms: \((2y-3)(y+2)\) and \((y+5)(y+2)\)These are binomials multiplied together and include a shared factor.
Polynomials can be classified according to:

• Degree: The highest power of the variable in the polynomial.
• Number of Terms: A polynomial with one term is a monomial. Two terms make it a binomial, three terms a trinomial, and so forth.

Understanding polynomials and recognizing their form is essential in algebra. It is crucial to know how to manipulate and simplify them by factoring.

Binomial Factor

A binomial factor is a polynomial with exactly two terms. In the context of the original exercise, factoring involves identifying and extracting a common binomial factor from polynomial expressions.Here, the common binomial factor is \((y+2)\),which appears in both terms of the polynomial expression, allowing it to be factored out entirely.The process involves:

• Identifying the common binomial factor present in each part of the expression.
• Factoring the binomial out, which involves writing the expression as a product of the binomial and a new expression derived from the terms.
• In the solution given, after factoring out \((y+2)\), the remaining expression is simplified to reach the final factorized form: \((y+2)(3y+2)\).

Recognizing and factoring binomial factors helps simplify polynomials and solve algebraic equations effectively.