Problem 37 - Elementary Algebra

Question

Factor. $$ 60 x 2-65 x+15 $$

Step-by-Step Solution

Answer

The factored form is \( 5(3x - 1)(4x - 3) \).

Step 1: Identify a Common Factor

First, observe the coefficients of the polynomial: 60, -65, and 15. Find the greatest common factor (GCF) among these numbers. The GCF is 5 as it divides all of them evenly. Factor 5 out of the expression.

Step 2: Factor Out the Greatest Common Factor

Divide each term in the polynomial by 5 and factor it out: \[ 5(12x^2 - 13x + 3) \]

Step 3: Identify the Inside Quadratic

Now focus on factoring the quadratic expression inside the parenthesis: \[ 12x^2 - 13x + 3 \]

Step 4: Find Two Numbers that Multiply to AC and Add to B

For the quadratic \( ax^2 + bx + c \), find two numbers that multiply to \( A imes C = 12 \times 3 = 36 \) and add to \( B = -13 \). These numbers are -9 and -4.

Step 5: Rewrite the Middle Term

Rewrite the quadratic by breaking the middle term using the numbers found: \[ 12x^2 - 9x - 4x + 3 \]

Step 6: Factor by Grouping

Group the terms into two pairs and factor each group: \[ (12x^2 - 9x) + (-4x + 3) \]The first group can be factored as \( 3x(4x - 3) \) and the second as \( -1(4x - 3) \).

Step 7: Complete the Factoring

Notice \( 4x - 3 \) is common in the groups. Factor \( 4x - 3 \) out:\[ (3x - 1)(4x - 3) \]

Step 8: Combine All Factors

Combine all the factors to get the fully factored form of the polynomial: \[ 5(3x - 1)(4x - 3) \]

Key Concepts

Greatest Common Factor

Understanding the Greatest Common Factor (GCF) is key to simplifying expressions. The GCF is the largest number that divides all terms in an expression without leaving a remainder. In the given polynomial, the coefficients are 60, -65, and 15. By identifying 5 as the GCF, we simplify our problem right from the start.

Here's how you determine it:

• Prime factorize each coefficient: 60 = 2² × 3 × 5, -65 = -1 × 5 × 13, and 15 = 3 × 5.
• The common factor across these numbers is 5.

By factoring out the GCF, we get: \[ 5(12x^2 - 13x + 3) \] This simplifies our task and makes further factoring easier.

Quadratic Expression

A quadratic expression is typically in the form of \( ax^2 + bx + c \). In our task, the expression inside the parentheses is \( 12x^2 - 13x + 3 \). Understanding the structure helps us factor it systematically.

This involves:

• Identifying coefficients: Here, \( a = 12 \), \( b = -13 \), and \( c = 3 \).
• Determining the product \( ac = 12 \times 3 = 36 \).
• Finding two numbers that multiply to 36 and sum to \( -13 \). These are -9 and -4.

Rewriting the expression using these numbers helps simplify it for factoring: \(12x^2 - 9x - 4x + 3\). Breaking it into these manageable parts sets up the next step: grouping.

Factoring by Grouping

Factoring by grouping breaks the expression into pairs that can be factored individually, making the problem approachable.

In our example, the focus is on \( 12x^2 - 9x - 4x + 3 \). Here's how you group:

• Group the terms: \((12x^2 - 9x) + (-4x + 3)\).
• Factor each group individually: \(3x(4x - 3) - 1(4x - 3)\).

The magic happens when you notice \(4x - 3\) is common in both parts. By factoring this term out, you simplify the expression to \((3x - 1)(4x - 3)\). This step effectively completes the process.

Combining it with the GCF, the full factorization becomes \[ 5(3x - 1)(4x - 3) \]. This results in a neatly factored form of the original polynomial.