Problem 57 - Algebra A Combined Function

Question

Factor each trinomial completely. See Examples 1 through 7. \(24 x^{2}-58 x+9\)

Step-by-Step Solution

Answer

The trinomial factors to \((6x - 1)(4x - 9)\).

Step 1: Identify Coefficients

For the trinomial \(24x^2 - 58x + 9\), identify the coefficients: \(a = 24\), \(b = -58\), and \(c = 9\).

Step 2: Multiply 'a' and 'c'

Calculate the product of \(a\) and \(c\): \(24 \times 9 = 216\).

Step 3: Find Two Numbers

Find two numbers whose product is \(216\) and whose sum is \(-58\). These numbers are \(-54\) and \(-4\), as \(-54 imes -4 = 216\) and \(-54 + (-4) = -58\).

Step 4: Rewrite Middle Term

Rewrite the trinomial \(24x^2 - 58x + 9\) as \(24x^2 - 54x - 4x + 9\) using the two numbers found in the previous step.

Step 5: Factor by Grouping

Group the terms: \((24x^2 - 54x) + (-4x + 9)\). Factor out the greatest common factor (GCF) from each group: \(6x(4x - 9) - 1(4x - 9)\).

Step 6: Factor Out Common Binomial

Factor out the common binomial \((4x - 9)\): \((6x - 1)(4x - 9)\).

Step 7: Verify Your Solution

Multiply the factors to ensure the product is the original trinomial. \((6x - 1)(4x - 9) = 24x^2 - 54x - 4x + 9 = 24x^2 - 58x + 9\). This confirms the factorization is correct.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra and appear as polynomial equations of degree two. They are generally written in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). In the given exercise, the quadratic equation is set as a trinomial on one side, \(24x^2 - 58x + 9\), which can be solved using multiple methods, including factoring. Understanding how to work with quadratic equations is vital because they pop up in various scenarios, from solving geometry problems to predicting business trends. The solutions to these equations can be real or complex numbers, depending on the discriminant \(b^2 - 4ac\). If you want to solve quadratics:

• Try to first set the equation to zero.
• Look for ways to factorize, if possible.
• Use the quadratic formula for more complex solutions.

When you factor these equations, you effectively split the quadratic into terms that are easier to solve.

Factor by Grouping

Factor by grouping is a great tool for breaking down complex expressions like polynomials into simpler parts. This method involves rearranging terms and finding common factors. For the trinomial \(24x^2 - 58x + 9\), the process began by reshuffling the middle term based on numbers that add to -58 and multiply to 216, the product of \(a\) and \(c\). With factor by grouping:

• First, rewrite the expression by splitting the middle term.
• Group the terms to show common factors.
• Factor out the greatest common factors from these groupings.
• Factor out the common binomial factor from the entire expression.

In our example, after the middle term split, the expression \(24x^2 - 54x - 4x + 9\) was grouped into \((24x^2 - 54x) + (-4x + 9)\). Each group had a common factor: \(6x\) for the first and -1 for the second. By factoring these, you reduce the expression. Finally, by factoring the common binomial \((4x - 9)\), you arrive at the factorized form \((6x - 1)(4x - 9)\).

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of simpler polynomials or factors. It is essential to simplify polynomial expressions and solve equations or graph polynomial functions efficiently. To understand polynomial factorization:

• Recognize that a polynomial is a sum of terms, each composed of a coefficient and variable raised to whole-number powers.
• The main goal is to express it in terms of products to reveal the roots or simplify expressions.
• This involves finding the greatest common factors, using techniques like factor by grouping or special formulas for quadratics and higher-order polynomials.

In the context of the given trinomial \(24x^2 - 58x + 9\), the factorization process reveals its roots by expressing it as the product of two binomial factors. These factors are easier to manipulate or solve, demonstrating the utility of polynomial factorization in simplifying complex expressions into manageable parts.