Problem 9 - Elementary Algebra

Question

Factor completely. $$ 5 x 3-30 x 2-15 x+90 $$

Step-by-Step Solution

Answer

The complete factorization is \(5(x^2 - 3)(x - 6)\).

Step 1: Identify Common Factors

Look at the expression \(5x^3 - 30x^2 - 15x + 90\). Identify the common factor in all terms, which is 5. Factor out 5: \(5(x^3 - 6x^2 - 3x + 18)\).

Step 2: Group Terms

Group the terms inside the parentheses to simplify further: \((x^3 - 6x^2) + (-3x + 18)\).

Step 3: Factor by Grouping

Factor out the greatest common factor from each group. For \((x^3 - 6x^2)\), factor out \(x^2\) to get \(x^2(x - 6)\). For \((-3x + 18)\), factor out \(-3\) to get \(-3(x - 6)\).

Step 4: Combine Common Factors

Notice both groups have a common factor of \((x - 6)\). Rewrite the expression: \((x^2 - 3)(x - 6)\).

Step 5: Write the Complete Factorization

Substitute back the common factor we removed in Step 1: \(5(x^2 - 3)(x - 6)\). This is the full factorization of the original polynomial.

Key Concepts

Factoring by Grouping

Factoring by grouping is a powerful tool when dealing with polynomial expressions, especially those with four or more terms. This method simplifies complex algebraic expressions, transforming them into products of simpler expressions. The key is to strategically group terms to reveal common factors.

• Start by identifying a common factor for each pair or group of terms.
• Extract this common factor, leaving you with a simpler expression within each group.
• Look for a common binomial factor across the grouped terms.

In our exercise, after identifying and factoring out the greatest factor from the entire expression, we group terms with similar variable powers. This makes it easier to handle the polynomial and find shared components that can be factored further. By doing this, you can transform complex polynomials into more manageable factors.

Algebraic Expressions

Understanding algebraic expressions is crucial, as these are the building blocks of algebra. They are composed of variables, coefficients, and arithmetic operations. Each part plays a distinct role in the formation of a polynomial.

• Variables: These are symbols, like \(x\) in our example, representing unknown values that can change.
• Coefficients: These are numbers multiplying the variables, such as 5, -30, and 18 in the equation.
• Terms: Each individual part of the expression separated by addition or subtraction, such as \(5x^3\) or \(-15x\).

Algebraic expressions come together to form equations and inequalities, which can be analyzed and factored for various mathematical applications. Our task is to break down these expressions into fundamental components, usually by using techniques like factoring to simplify or solve them.

Greatest Common Factor

The greatest common factor (GCF) is the largest number or expression that can divide all terms in an algebraic expression evenly. It serves as an essential step in the simplification process, especially before utilizing methods like factoring by grouping.

• The GCF helps in minimizing the complexity of the polynomial, making further factorization easier.
• It involves looking at the coefficients and variables' powers common across all terms.

In the polynomial \(5x^3 - 30x^2 - 15x + 90\), the GCF of 5 is factored out first. This extractable factor streamlines subsequent operations, enabling us to proceed with factoring by grouping. Identifying the GCF is often the first step in making the polynomial more manageable and setting the stage for detailed analysis or simplification.