Problem 28 - Algebra A Combined Function

Question

Factor each trinomial completely. See Examples 1 through 5 . \(-3 t+4 t^{2}-7\)

Step-by-Step Solution

Answer

The trinomial \(-3t + 4t^2 - 7\) factors as \((t + 1)(4t - 7)\).

Step 1: Rearrange the Trinomial

First, rewrite the trinomial in standard form. The standard form for a polynomial is to arrange it in descending powers of its variable. The trinomial given is \(-3t + 4t^2 - 7\). Rearrange it to get \(4t^2 - 3t - 7\).

Step 2: Identify the Coefficients

Identify the coefficients for the quadratic expression \(4t^2 - 3t - 7\), where the quadratic is in the form \(ax^2 + bx + c\). Here, \(a = 4\), \(b = -3\), and \(c = -7\).

Step 3: Use the AC Method

Since the leading coefficient \(a\) is not 1, use the AC method for factoring. Multiply \(a\) and \(c\) to find \(ac = 4 \, * \, (-7) = -28\). Find two integers that multiply to \(-28\) and add to \(b = -3\). The integers \(4\) and \(-7\) satisfy this, as \(4 - 7 = -3\).

Step 4: Rewrite the Middle Term

Rewrite the middle term \(-3t\) using the two integers found: \(4t^2 + 4t - 7t - 7\).

Step 5: Factor by Grouping

Group the terms in pairs and factor each pair separately: \((4t^2 + 4t) + (-7t - 7)\). Factor out \(4t\) from the first group: \(4t(t + 1)\). Factor \(-7\) from the second group: \(-7(t + 1)\). This gives \(4t(t + 1) - 7(t + 1)\).

Step 6: Factor Out the Common Binomial

Notice the common binomial factor is \((t + 1)\). Factor this out to get: \((t + 1)(4t - 7)\). Hence, the trinomial \(4t^2 - 3t - 7\) is factored as \((t + 1)(4t - 7)\).

Key Concepts

Trinomial Factorization

Trinomial factorization is an essential skill when working with polynomial expressions. A trinomial is simply a polynomial with three terms. Factoring it means breaking it down into simpler expressions, usually the product of two binomials.
This technique is particularly useful for solving quadratic equations, as it helps identify the roots or solutions of the equation. The goal is to express the trinomial, typically in the form \(ax^2 + bx + c\), as a product of two binomials \((dx + e)(fx + g)\).
The process often begins with rearranging the polynomial, if necessary, to ensure terms are in descending order of their powers. The coefficients for each term are identified, as they guide the next factorization steps.

AC Method

The AC method is a systematic approach to factor trinomials when the leading coefficient \(a\) is not equal to one. This method relies on multiplying \(a\) and \(c\) to find a product \(ac\) that helps determine the integer pairs needed for factorization.
The key steps include:

• Multiply the leading coefficient \(a\) by the constant term \(c\) to find \(ac\). For instance, in the trinomial \(4t^2 - 3t - 7\), \(ac = 4 \, * \, (-7) = -28\).
• Find two integers that multiply to \(ac\) and add to the middle term's coefficient \(b\). In this case, integers 4 and -7 satisfy the requirement because their product is -28 and their sum is -3.
• Once these integers are found, the original trinomial can be rewritten to aid in factorization through grouping.

This structured methodology makes it easier to factorize even complex trinomials with greater confidence.

Grouping Method

The grouping method is a clever way to factor a polynomial by rearranging and grouping terms to reveal common factors. It's particularly useful after rewriting the middle term with the integers found through the AC method.
Here's a concise breakdown of how grouping is applied:

• Separate the trinomial into two groups based on the integers found. Using our example, rewrite \(4t^2 - 3t - 7\) as \((4t^2 + 4t) + (-7t - 7)\).
• Factor out the greatest common factor from each group: \(4t(t + 1)\) from the first group and \(-7(t + 1)\) from the second.
• Once each group is factored, look for a common binomial factor, here \((t + 1)\), that can be factored out from both groups.

This method can simplify polynomials and is particularly powerful in conjunction with the AC method.

Polynomial Expressions

Polynomial expressions consist of variables raised to various powers and coefficients, such as \(4t^2 - 3t - 7\). These mathematical expressions are foundational to algebra and encompass a wide range of operations and transformations.
Understanding polynomial expressions involves recognizing:

• The terms: Each separate part of the polynomial, such as \(4t^2\), \(-3t\), and \(-7\).
• The degree of the polynomial, which is the highest power of the variable present—in this case, degree 2.
• How different operations apply to them, like addition, subtraction, multiplication, and, importantly, factoring.

Factoring is a key operation for simplifying polynomial expressions and solving polynomial equations. It breaks down complicated expressions into products of simpler polynomials, often revealing roots or intersections.