Problem 1 - Algebra and Trigonometry

Question

1–8 ? Factor out the common factor. $$ 5 a-20 $$

Step-by-Step Solution

Answer

The expression factors to \(5(a - 4)\).

Step 1: Identify the Common Factor

Look at the terms in the expression \(5a - 20\). Identify any number or variable that is a factor of both terms. In this case, the number 5 is a common factor because both 5 and 20 are divisible by 5.

Step 2: Factor Out the Common Factor

Take the common factor (5) out of the expression. Divide each term by 5. Rewrite the expression with the common factor factored out:\[ 5(a - 4) \]Here, \(5a \div 5 = a\) and \(-20 \div 5 = -4\).

Step 3: Verify the Factored Expression

Multiply the factored expression back to check its correctness:\(5(a - 4) = 5 \cdot a - 5 \cdot 4 = 5a - 20\).This confirms that the factored form is correct.

Key Concepts

Common Factor

When you begin factoring algebraic expressions, the first task is to identify the common factor. A common factor is a number or variable that divides all the terms in an expression without leaving a remainder. For instance, in the expression \(5a - 20\), the common factor is 5 because both terms 5 and 20 can be evenly divided by 5. Finding the common factor is crucial because it simplifies the expression and serves as the foundation for further factorization steps.

• Look at all the numerical coefficients of the terms.
• Find the largest number that divides each coefficient evenly.
• Check if any variables are common across all terms.
• Combine these values to find the overall common factor.

Understanding and identifying common factors is an essential skill that will make algebra much easier to manage.

Factored Expression

Once you have identified the common factor, the next step is to rewrite the expression in its factored form. A factored expression is simply the expression rewritten as a product, using the common factor.
For the expression \(5a - 20\):
- We divide each term by the common factor, which is 5. This simplifies to \(a - 4\).
- Write the expression as a multiplication of the common factor and this simplified expression, giving: \(5(a - 4)\).

Factored expressions are useful because they reveal the underlying structure of an algebraic expression, making further manipulation, such as solving equations or integration, smoother. Always remember to double-check by distributing the common factor back into the factored terms to ensure accuracy.

Algebraic Factorization

Algebraic factorization is a technique used to break down complex expressions into simpler components. It’s like finding pieces that multiply together to give the original expression. This process is powerful because it can help simplify calculations and solve equations more efficiently.

• Algebraic factorization begins by identifying a common factor, just as we did with \(5a - 20\).
• The expression is then rewritten using this factor, making it easier to work with.
• More complex expressions might involve multiple steps of finding and factoring common factors.

Through algebraic factorization, you can simplify expressions, identify roots of equations, and even work with polynomials more effectively. Understanding this concept opens up many paths in algebra and is foundational for higher mathematics.