Problem 11 - A Graphical Approach to College Algebra

Question

Factor the greatest common factor from each polynomial. $$2(m-1)-3(m-1)^{2}+2(m-1)^{3}$$

Step-by-Step Solution

Answer

The factored form is \((m-1)(2m^2 - 7m + 7)\).

Step 1: Identify the common factor

Observe the polynomial \(2(m-1) - 3(m-1)^2 + 2(m-1)^3\). Notice that each term includes the expression \((m-1)\). We will factor out the greatest common factor which is \((m-1)\).

Step 2: Factor out the greatest common factor \((m-1)\)

Extract \((m-1)\) from each term in the polynomial. This gives us:\[(m-1)(2 - 3(m-1) + 2(m-1)^2)\].

Step 3: Simplify the expression inside the parentheses

Now, simplify the expression inside the parentheses:1. Expand \(-3(m-1)\) to get \(-3m + 3\).2. Expand \(2(m-1)^2\) to get \(2(m^2 - 2m + 1) = 2m^2 - 4m + 2\).3. Combine like terms: the expression becomes \(2 - 3m + 3 + 2m^2 - 4m + 2) = 2m^2 - 7m + 7\).

Step 4: Write the complete factored form

After simplification, the factored expression is:\[(m-1)(2m^2 - 7m + 7)\].

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is the largest expression that is a factor of all terms in a polynomial. It allows us to simplify the polynomial by factoring it out. In the example given, the expression \((m-1)\) is present in each term of the polynomial. Therefore, \((m-1)\) is the GCF. Finding and factoring the GCF:

• Identify the repeated expression in each term.
• Factor it out from the entire polynomial.

By factoring \((m-1)\) from the expression \(2(m-1) - 3(m-1)^2 + 2(m-1)^3\), we simplify the problem and reveal underlying patterns.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This process often includes expanding and combining like terms to ease understanding and further calculations.

Step-by-Step Simplification

To simplify the expression inside the parentheses \( (2 - 3(m-1) + 2(m-1)^2) \):

• Start by expanding each term: \(-3(m-1)\) becomes \(-3m + 3\), and \(2(m-1)^2\) becomes \(2(m^2 - 2m + 1) = 2m^2 - 4m + 2\).
• Next, combine like terms, which are terms with the same variables raised to the same power: \(2 - 3m + 3 + 2m^2 - 4m + 2\) simplifies to \(2m^2 - 7m + 7\).

Through these steps, the expression becomes more manageable and clear.

Polynomial Expressions

Polynomial expressions are mathematical phrases that can have constants, variables, and exponents combined using addition, subtraction, and multiplication.

Understanding the Structure

In our example, the original expression \(2(m-1) - 3(m-1)^2 + 2(m-1)^3\) consists of multiple polynomials and factors.

Key Characteristics

• Degree: The highest power of the variable \(m\) determines the degree of the polynomial.
• Terms: Each part of the expression separated by a plus or minus sign.
• Coefficients: The numbers in front of the variables.

Polynomial expressions play a crucial role in algebra, providing a foundation for more complex mathematical concepts and operations.