Problem 21 - A Graphical Approach to College Algebra

Question

Factor each trinomial completely. $$8 h^{2}-24 h-320$$

Step-by-Step Solution

Answer

The complete factorization is \(8(h - 8)(h + 5)\).

Step 1: Identify a common factor

First, identify if there is a common factor in all terms of the trinomial. In this case, all coefficients are divisible by 8. So, the common factor is 8.

Step 2: Factor out the common factor

Divide each term of the trinomial by 8 and factor it out: \[ 8(h^2 - 3h - 40) \]

Step 3: Factor the quadratic trinomial

Now, focus on the trinomial inside the parentheses: \( h^2 - 3h - 40 \). We need to factor this into two binomials. We look for two numbers that multiply to -40 and add to -3.

Step 4: Find the numbers

The numbers that multiply to -40 and add up to -3 are -8 and 5. This is because \(-8 \times 5 = -40\) and \(-8 + 5 = -3\).

Step 5: Write the binomial factors

Using the numbers found in the previous step, rewrite the trinomial as two binomials:\[ (h - 8)(h + 5) \]

Step 6: Combine all factors

Combine the common factor from Step 2 with the binomial factors from Step 5:\[ 8(h - 8)(h + 5) \]

Step 7: Verify the factorization

Expand \(8(h - 8)(h + 5)\) to ensure it equals the original expression. Doing this multiplication will confirm the factorization is correct.

Key Concepts

Common Factor

Factoring trinomials often begins with identifying a common factor in all terms. This step simplifies the process significantly by reducing the entire expression. Think of it as "decluttering" before organizing. In the example given, 8 is the common factor. Here’s how you check for a common factor:

• Look at the coefficients of all terms in the trinomial.
• Find the greatest number that evenly divides each of them.

Once identified, simply factor it out by dividing each term in the trinomial by this number. Consequently, you arrive at a simpler expression to work with. By focusing on common factors first, the subsequent steps become more manageable, ensuring you maintain simplicity throughout each stage.

Quadratic Trinomial

Delving into the heart of factoring trinomials, you must understand what a quadratic trinomial is. It is an expression of the form: \[ ax^2 + bx + c \]Where \(a\), \(b\), and \(c\) are constants. The critical challenge is to split it into two binomial expressions. In the problem provided, the goal is to express \(h^2 - 3h - 40\) in such a manner. The steps taken include:

• Finding two numbers that multiply to the product of \(a \times c\) (from the quadratic) and add to \(b\) (the coefficient of the middle term).
• Using these numbers to split the middle term and rewrite the quadratic trinomial accordingly.

Understanding these foundational aspects helps to tackle the trinomial with a structured approach, ensuring clarity and avoiding errors.

Binomial Factors

The culmination of factoring a quadratic trinomial is breaking it down into binomial factors. These binomials are expressions of the form \((x + m)(x + n)\), representing a product of two simpler expressions derived from the original equation.In the given exercise, the trinomial \(h^2 - 3h - 40\) was decomposed into the binomials \((h - 8)(h + 5)\). Here is how this was achieved:

• Using numbers identified earlier (-8 and 5) to rewrite the trinomial.
• Assemble these numbers into binomials such that their factors align with our requirement (multiply to the constant term, add to the linear term coefficient).

Finally, always remember to verify your results by expanding the expressions. This ensures that the factorization returns to the original trinomial. This step safeguards your calculations, providing assurance that the factored form is correct.