Problem 89 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x+c\). \(w^{2}-5 w-36\)

Step-by-Step Solution

Answer

\((w + 4)(w - 9)\)

Step 1: Identify coefficients

Identify the coefficients from the trinomial in the form x^2 + bx + c. In this case, we have b = -5 and c = -36.

Step 2: Find factor pairs of c

List all factor pairs of the constant term c (-36). These are: (1, -36), (-1, 36), (2, -18), (-2, 18), (3, -12), (-3, 12), (4, -9), (-4, 9), (6, -6), and (-6, 6).

Step 3: Determine the correct factor pair

Find the pair of factors that add up to b (-5). The correct pair from our list is (4, -9) because 4 + (-9) = -5.

Step 4: Write the factors

Rewrite the trinomial as a product of two binomials using the factor pair (4, -9). \((w + 4)(w - 9) \).

Step 5: Confirm the factorization

Expand the binomials to ensure they multiply back to the original trinomial: \((w + 4)(w - 9) = w^2 - 9w + 4w - 36 = w^2 - 5w - 36 \)

Key Concepts

Identifying Coefficients

To factor a trinomial, the first task is to identify its coefficients. The coefficients are the numerical parts of each term in the algebraic expression. In our trinomial, which is in the form \(w^2 - 5w - 36\), the terms are:

• The coefficient of \(w^2\) (the quadratic term) is 1. This is because any variable raised to a power without a visible coefficient is assumed to have a coefficient of 1.
• The coefficient of \(w\) (the linear term) is -5.
• The constant term, which isn’t multiplied by any variable, is -36.

Knowing these coefficients helps us understand the structure of the trinomial and is crucial for the next steps in the factoring process.

Factor Pairs

Once we've identified our coefficients, we need to find factor pairs of the constant term. The factor pairs of -36 are combinations of numbers that multiply to -36. They are:

• (1, -36)
• (-1, 36)
• (2, -18)
• (-2, 18)
• (3, -12)
• (-3, 12)
• (4, -9)
• (-4, 9)

We list these factor pairs because one of these pairs will not only multiply to -36 but also add up to our b coefficient, which is -5. This particular characteristic will help identify the correct pair to use in our factoring.

Binomials

After identifying the correct factor pair, we use them to write our trinomial as a product of two binomials. In our case, the correct pair is (4, -9). This means if we split the middle term, -5w, using these factors, we can rewrite the original trinomial. Our trinomials will then look like this:
\((w + 4)(w - 9)\).
To verify, expand this binomial expression by using the distributive property:

• \((w + 4)(w - 9) = w(w) + w(-9) + 4(w) + 4(-9)\)
• \(w^2 - 9w + 4w - 36 = w^2 - 5w - 36\)

This confirmation shows that our factorization is correct.

Algebraic Expressions

Trinomials are a type of algebraic expression that contains three terms. These terms are connected by addition or subtraction. Factoring trinomials involves rewriting them as a product of simpler algebraic expressions (binomials). This makes it easier to solve equations or simplify expressions. In algebra, an essential skill is manipulating these expressions to find simpler forms or solve for a variable. By breaking down a trinomial into binomials, we make the expression more manageable.
Understanding how to identify coefficients, find correct factor pairs, and build binomials are the foundational steps in mastering algebraic expressions.