Problem 5 - College Algebra with Corequisite Support

Question

For the following exercises, find the greatest common factor. \(49 m b^{2}-35 m^{2} b a+77 m a^{2}\)

Step-by-Step Solution

Answer

The greatest common factor is \(7m\).

Step 1: Identify the coefficients and variable parts

In the expression \(49mb^{2} - 35m^{2}ba + 77ma^{2}\), we need to find the greatest common factor (GCF) by examining each term. The coefficients are 49, -35, and 77. The variable parts include powers of \(m\), \(b\), and \(a\).

Step 2: Find the GCF of the numerical coefficients

Identify the numerical coefficients: 49, 35, and 77. Find the GCF of these numbers by determining their factors. The prime factors for 49 are \(7 \times 7\), for 35 are \(5 \times 7\), and for 77 are \(7 \times 11\). The common prime factor is 7, so the GCF of the coefficients is 7.

Step 3: Find the GCF of the variable parts

For the variable parts, examine each part separately:- \(m^1, m^2, m^1\) has a minimum power of \(m^1\).- \(b^2, b^1, b^0\) has a minimum power of \(b^0\) .- \(a^0, a^1, a^2\) has a minimum power of \(a^0\) .The GCF of the variable parts is \(m\).

Step 4: Combine the results

Combine the results from Steps 2 and 3. The GCF for the entire expression is the product of the numerical GCF and the variable GCF. Therefore, the GCF is \(7m\).

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. In the expression \(49mb^{2} - 35m^{2}ba + 77ma^{2}\), each part, separated by a plus or minus sign, is called a term. Algebraic expressions can have one or more terms, and they may include:

• Numerical coefficients: These are the numbers that multiply the variables in each term. For example, in the term \(49mb^{2}\), 49 is the coefficient.
• Variables: Symbols like \(m\), \(b\), and \(a\) that represent unknown values. They can appear in different powers within the expression.
• Operations: Such as addition \((+)\) and subtraction \((-).\)

Understanding the structure of algebraic expressions is crucial for simplifying them or finding common factors, as seen in our example.

Prime Factorization

Prime factorization involves breaking down a number into its basic building blocks or prime numbers, which are numbers greater than 1 that have no divisors other than 1 and themselves. For instance, the prime factorization of 49 is \(7 \times 7\), since 7 is a prime number. Likewise, the number 35 is expressed as \(5 \times 7\), and 77 as \(7 \times 11\).

This process is beneficial in algebra for finding the greatest common factor (GCF) of numerical coefficients. By identifying all the prime factors of each coefficient, we can determine the largest factor they share, which in this case is 7.

• Useful for simplifying expressions by reducing them to their simplest form.
• Helps in solving equations where common factors can be factored out.

Variable Powers

In algebra, variables may appear with different powers, indicating how many times a variable is multiplied by itself. In our example, powers such as \(b^{2}\), \(b^{1}\), and \(b^{0}\) suggest the exponentiation levels of variable \(b\).

• A power of 1, such as \(m^{1}\), means the variable is just \(m\).
• A power of 0, such as \(b^{0}\) or \(a^{0}\), simplifies to 1 because any number to the power of zero is one.

Understanding variable powers helps in identifying the greatest common factor among variables. Here, the lowest powers of each variable in all the terms were chosen to determine the GCF as \(m\), since it's present in all terms with at least power one.

Polynomials

Polynomials are a special type of algebraic expressions that consist of multiple terms. Each term in a polynomial is made up of a coefficient, variables, and exponents. The expression \(49mb^{2} - 35m^{2}ba + 77ma^{2}\) is a polynomial consisting of three distinct terms.

Polynomials can be classified based on the number of terms they have:

• Monomial: A polynomial with a single term, like \(7x^2\).
• Binomial: A polynomial with two terms, like \(x+2\).
• Trinomial: A polynomial with three terms, like our example expression.

Polynomials can be added, subtracted, multiplied, and divided, making them very versatile in algebra. Understanding polynomials and their properties is crucial for breaking down expressions to their simplest forms, solving equations, and factoring out terms, as shown in the calculation of the greatest common factor.