Problem 57 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(4 m^{3}+10 m^{2}-6 m\)

Step-by-Step Solution

Answer

The fully factored expression is \(2m(2m - 1)(m + 3)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, look at the coefficients (4, 10, and -6) and the variable powers in each term of the polynomial. We determine that each term has an 'm' and can be factored out. The GCF of the coefficients 4, 10, and -6 is 2. So, the GCF of the whole expression is 2m.

Step 2: Factor Out the GCF

Once you have identified the GCF (2m), divide each term of the polynomial by 2m and take it outside of the parentheses. This gives: \(2m(2m^2 + 5m - 3)\).

Step 3: Factor the Quadratic Expression

Now, look inside the parentheses at the quadratic expression \(2m^2 + 5m - 3\). To factor it, we need two numbers that multiply to \(-6\) (the product of the leading coefficient 2 and the constant term -3) and add up to 5 (the middle coefficient).

Step 4: Find Factor Pair

The numbers are 6 and -1 because 6 * -1 = -6 and 6 + (-1) = 5. We use these numbers to rewrite the middle term: \(2m^2 + 6m - m - 3\).

Step 5: Perform Grouping

Group the terms into two pairs: \((2m^2 + 6m)\) and \((-m - 3)\). Factor out the common factors from each pair: \(2m(m+3) - 1(m+3)\).

Step 6: Factor by Grouping Combined Expression

Notice that \((m+3)\) is a common factor in both groups, so factor it out: \((2m - 1)(m + 3)\).

Step 7: Final Expression and Verification

The expression is completely factored as \(2m(2m - 1)(m + 3)\). To verify, expand the expression to check it matches the original: \[2m(2m^2 + 5m - 3) = 4m^3 + 10m^2 - 6m\], confirming the factorization is correct.

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) of a polynomial is the largest quantity that divides each term of the polynomial without leaving a remainder. To find the GCF of a polynomial like \(4m^3 + 10m^2 - 6m\), first, consider the coefficients: 4, 10, and -6. The greatest common factor of these numbers is 2, as it is the largest number that can divide each one evenly.
Next, observe the variable powers. All terms contain the variable \(m\), and the smallest power of \(m\) in the expression is \(m^1\). Thus, the GCF of the entire expression is \(2m\).

• Step 1: Identify the GCF among coefficients and variables.
• Step 2: Use the GCF to factor out from each term.

Utilizing the GCF makes the expression easier to manage by reducing its complexity and revealing other factors further in the simplification process.

Quadratic Expression

Quadratic expressions are polynomials of the form \(ax^2 + bx + c\). These expressions can often be factored into simpler binomials if they meet certain criteria. In the context of the polynomial \(4m^3 + 10m^2 - 6m\), once the GCF, \(2m\), is factored out, you are left with a simpler quadratic: \(2m^2 + 5m - 3\).
When dealing with quadratic expressions, the typical approach is to find two numbers that multiply to \(a \times c\) (the product of the first and third term coefficients) and add to \(b\) (the middle coefficient).

• The quadratic can then be factored further using various methods, like the grouping method.
• This simplifies the work necessary to completely factor a given polynomial.

Mastering the factorization of quadratic expressions is key to simplifying complicated polynomials.

Factor Pair

The process of finding two numbers that multiply to a specific product and add up to a specific sum is crucial in factorization. In the expression \(2m^2 + 5m - 3\), derived in the factorization steps from \(4m^3 + 10m^2 - 6m\), we need a factor pair for the product \(-6\) (from \(2 \times -3\)) that sums to 5.
The suitable factor pair here is 6 and -1:

• 6 and -1 multiply to \(-6\): \(6 \times (-1) = -6\).
• 6 and -1 add up to 5: \(6 + (-1) = 5\).

By using these numbers, you rewrite the middle term \(5m\) in the quadratic expression as \(6m - m\). This reveals an opportunity to easily apply methods such as grouping to finalize the factorization.
Finding the correct factor pair is an integral step when dealing with quadratic expressions to progress effectively in simplifying polynomials.

Polynomials Grouping

Grouping is a technique used in the factorization of polynomials to organize terms in such a way that allows common factors to appear. After splitting the quadratic expression \(2m^2 + 5m - 3\) into \(2m^2 + 6m - m - 3\) using the factor pair method, you can see two groups: \((2m^2 + 6m)\) and \((-m - 3)\).
What you do next is take the greatest common factor out of each group separately:

• In the first group \((2m^2 + 6m)\), factor out \(2m\), resulting in \(2m(m + 3)\).
• In the second group \((-m - 3)\), factor out \(-1\), giving \(-1(m + 3)\).

Once both groups have been factored, you notice that \((m + 3)\) is common to both. This common factor can be factored out:
\[(2m(m + 3) - 1(m + 3)) = (2m - 1)(m + 3)\].
Grouping and then factoring a common term from each group helps to simplify polynomial expressions efficiently, ensuring complete factorization of the problem.