Problem 46 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 28-3 m-m^{2} $$

Step-by-Step Solution

Answer

The expression factors to \(-(m - 4)(m + 7)\).

Step 1: Identify the Expression

The given expression is \(28 - 3m - m^2\). This is a quadratic expression that can be rewritten in standard form as \(-m^2 - 3m + 28\).

Step 2: Reorder into Standard Form

Rearrange the terms so the expression is in the standard quadratic form \(ax^2 + bx + c\). The expression becomes \(-m^2 - 3m + 28\).

Step 3: Check for Common Factors

Examine the expression \(-m^2 - 3m + 28\) for any common factors in all the terms. In this case, there are no common factors to factor out.

Step 4: Use Factoring Techniques

Attempt to factor the quadratic. For an expression of the form \(-m^2 - 3m + 28\), let's rewrite it as \(-(m^2 + 3m - 28)\). We factor the expression inside the parentheses.

Step 5: Factor the Quadratic

We need two numbers which multiply to \(-28\) (the product of \(1\) and \(-28\)) and add to \(3\). The numbers \(7\) and \(-4\) satisfy this. The expression becomes \(-((m - 4)(m + 7))\).

Step 6: Simplify the Factorized Expression

Remove the negative sign or leave it intact as per required expression stylizing: \(-(m - 4)(m + 7)\) is generally accepted as correct.

Key Concepts

Standard Quadratic Form

Quadratic expressions need to be in standard form to be factored easily. The standard quadratic form is written as \( ax^2 + bx + c \).
Here, \( a \) is the coefficient of \( x^2 \), \( b \) is the coefficient of \( x \), and \( c \) is the constant term.
For example, in the expression \( -m^2 - 3m + 28 \), the coefficients are \( a = -1 \), \( b = -3 \), and \( c = 28 \).
Rearranging expressions into this form can make it easier to identify how to factor them. Keeping the terms in a consistent order helps apply various factoring methods more effectively. This standard form is particularly useful for visualizing the relationships between the terms, thereby aiding in the factoring process.

Common Factors

Before diving into more complex factoring techniques, it's important to first check for any common factors in all terms of a polynomial expression.
A common factor is a number or variable that divides exactly into each term in the expression.
For example, in the expression \(-m^2 - 3m + 28\), you initially check to see if there is a common factor for all the terms: \(-m^2\), \(-3m\), and \(28\). In this case, there are no common factors.
Identifying common factors early can simplify the expression before applying more complex factoring techniques. It makes the latter steps cleaner and often more straightforward. Being diligent with this step will save time and prevent errors in more advanced processes.

Factoring Techniques

Once it’s confirmed that no common factors are present, various factoring techniques come into play.
A popular method is to look for two numbers that multiply to give the product of \( ac \) (in standard form \( ax^2 + bx + c \)), and add to give \( b \).
In the expression \(-m^2 - 3m + 28\), we look for two numbers that multiply to \(-28\) and add to \(3\). The numbers \(7\) and \(-4\) work perfectly because \(7 \cdot (-4) = -28\) and \(7 + (-4) = 3\).
Thus, we can rewrite the expression as \( -(m - 4)(m + 7) \). Using these numbers to split the middle term leads to an easier pathway to factor the entire quadratic. Explore different combinations systematically to find the pair that satisfies the conditions.

Simplifying Expressions

Simplifying expressions is crucial in making mathematical expressions more manageable and easier to interpret.
After factoring, expressions often look like long chains of numbers and variables multiplied together.
In our expression \(-(m - 4)(m + 7)\), simplification makes it more presentable and feasible to work with further.
Decide whether to leave the negative sign outside or distribute it across one of the factors, based on the context. The simplified form can be left as \(-(m - 4)(m + 7)\) since it is an acceptable formatted factorization. Remember, properly simplified expressions make calculations easier, reduce errors, and are generally aesthetically pleasing.