Problem 37 - Elementary and Intermediate Algebra

Question

Factor. See Example 3 or Example 6. $$ 5 m^{2}-7 m-6 $$

Step-by-Step Solution

Answer

The factored form of \(5m^2 - 7m - 6\) is \((5m + 3)(m - 2)\).

Step 1: Identify the coefficients

The given quadratic expression is \( 5m^2 - 7m - 6 \). Here, the coefficients are: - \(a = 5\) for \(m^2\), - \(b = -7\) for \(m\), - \(c = -6\)

Step 2: Find two numbers that multiply and add/subtract to get known values

We need two numbers that multiply to \(a \cdot c\) (which is \(5 \times -6 = -30\)) and add to \(b\) (which is \(-7\)). After testing several pairs, we find that \(-10\) and \(3\) multiply to \(-30\) and add to \(-7\).

Step 3: Rewrite the expression by splitting the middle term

Using the numbers found in Step 2, rewrite the middle term \(-7m\) as \(-10m + 3m\): \[ 5m^2 - 10m + 3m - 6 \]

Step 4: Group terms and factor by grouping

Group the terms into two pairs and factor each group: \[ (5m^2 - 10m) + (3m - 6) \]Factor out the greatest common factor from each group: \[ 5m(m - 2) + 3(m - 2) \]

Step 5: Factor out the common binomial

The expression now can be factored by taking common the binomial \((m - 2)\), which results in: \[ (5m + 3)(m - 2) \]

Key Concepts

Understanding Quadratic Expressions

A quadratic expression is a type of polynomial that includes a variable raised to the power of two. It takes the form:

• \( ax^2 + bx + c \)

Where:

• \( a \), \( b \), and \( c \) are constants.
• \( x \) is the variable.

It's called "quadratic" because the highest exponent of the variable is 2. These expressions form a parabola when graphed, which is a curved, U-shaped line. Quadratic expressions are widely encountered in algebra, and knowing how to manipulate them is crucial for solving many types of mathematical problems.

Coefficient Identification

Identifying coefficients in a quadratic expression is the first step in the factoring process. A coefficient is a numerical or constant quantity placed before and multiplying the variable in an algebraic term.

In the quadratic expression \( ax^2 + bx + c \):

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

In our example, \( 5m^2 - 7m - 6 \), we identify:

• \( a = 5 \)
• \( b = -7 \)
• \( c = -6 \)

Accurate identification of these coefficients is vital. It helps us construct the equation and manipulate it in the subsequent steps for factoring.

Factoring by Grouping

Factoring by grouping is a useful technique when attempting to factor quadratic expressions that don't have a straightforward way of splitting.

Once you've rewritten the middle term, the aim is to group terms in pairs, which can be factored separately. Here's a brief breakdown of this process:

• First, rewrite the expression by splitting the middle term.
• Then, group terms in pairs that have a common factor.
• Factor out the greatest common factor from each pair.

For the expression \( 5m^2 - 10m + 3m - 6 \), we group as:

• \( 5m^2 - 10m \)
• \( 3m - 6 \)

Next, factor out the common factor from each:

• \( 5m(m - 2) \)
• \( 3(m - 2) \)

This method effectively simplifies the expression, setting the stage for binomial factorization.

The Binomial Factorization Method

After factoring by grouping, the final task is the binomial factorization process. This involves extracting a common binomial factor from the expression.

In our case, the previously grouped pairs lead to:

• \( 5m(m - 2) + 3(m - 2) \)

Notice that \((m - 2)\) appears in both terms. We can factor this binomial out:

• \((5m + 3)(m - 2)\)

Congratulations! We've completed the factoring process by identifying and extracting the common binomial factor, simplifying the original quadratic expression into a product of two binomials. This skill is foundational in algebra, facilitating the solving of equations and analysis of expressions.