Problem 102 - College Algebra with Modeling and Visualization

Question

Factor the expression. \(5 t^{6}+40 r^{3}\)

Step-by-Step Solution

Answer

Factored expression: \(5(t^6 + 8r^3)\).

Step 1: Identify Common Factor

The given expression is \(5t^6 + 40r^3\). Notice that both terms share the common factor of 5. Extracting 5 as the greatest common factor (GCF) simplifies the expression.

Step 2: Factor Out the GCF

Factor 5 out of each term of the expression: \(5t^6 + 40r^3 = 5(t^6 + 8r^3)\). This simplifies the expression by removing the common factor.

Step 3: Analyze Remaining Expression

After factoring out the GCF, we are left with \(t^6 + 8r^3\). Neither term in this expression shares any other common factor besides 1, so no further factorization is needed.

Step 4: Conclusion: Write the Factored Expression

The complete factored form of the original expression is \(5(t^6 + 8r^3)\). This is the simplest form of the expression using factoring.

Key Concepts

Greatest Common Factor (GCF)

The greatest common factor, or GCF, is the largest factor that divides two or more numbers or terms. When dealing with algebraic expressions, finding the GCF is a crucial step in the factoring process.

In the original exercise, we consider the expression with two terms: \(5t^6 + 40r^3\).

• The number 5 is a common factor of both 5 and 40.
• No variables are shared, so the GCF is simply the numerical factor.

Finding the GCF helps in simplifying expressions by extracting this factor and reducing the complexity of the terms involved.

Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients, constructed using operations of addition, subtraction, and multiplication.

The expression \(5t^6 + 40r^3\) is a polynomial.

• Each term in a polynomial is made up of a constant multiplier and a variable raised to a power, known as a monomial. Here, \(5t^6\) and \(40r^3\) are monomials.
• The polynomial degree is determined by the highest power of the variable in the expression. In this case, the degree of each term determines how complex the polynomial is, but for factoring, we're mostly concerned with the common factors of the terms.

Recognizing that an expression is a polynomial allows us to apply a variety of algebraic manipulations, such as factoring.

Algebraic Expressions

Algebraic expressions are combinations of numbers, symbols, and operators representing quantities.

Expressions like \(5t^6 + 40r^3\) are fundamental constructs in algebra which follow specific rules for simplification and manipulation.

• They can be simplified or combined using basic arithmetic and algebraic factoring, such as finding the GCF.
• The expression is not an equation since it doesn’t include an equal sign, but a collection of terms we can manipulate for simplicity or depending on the problem's needs.

Understanding algebraic expressions is key to solving equations, simplifying them, and accurately interpreting mathematical problems.