Problem 19 - Algebra A Combined Function

Question

Factor each trinomial completely. If a polynomial can't be factored, write "prime." $$ 10 t-24+t^{2} $$

Step-by-Step Solution

Answer

The trinomial factors to \((t + 12)(t - 2)\).

Step 1: Reorder the Polynomial

First, rearrange the given trinomial in standard form as follows: \[ t^2 + 10t - 24 \] where the terms are ordered by decreasing powers of \( t \).

Step 2: Identify Coefficients

Identify the coefficients of the trinomial: \( a = 1 \) (coefficient of \( t^2 \)), \( b = 10 \) (coefficient of \( t \)), and \( c = -24 \) (constant term).

Step 3: Factor Using AC Method

For the trinomial \( t^2 + 10t - 24 \), calculate \( a \times c = 1 \times (-24) = -24 \). We need factors of \(-24\) that add up to \(10\). The factors \(12\) and \(-2\) satisfy this requirement.

Step 4: Rewrite the Middle Term

Rewrite the trinomial using the chosen factors: \[ t^2 + 12t - 2t - 24 \]. This allows the trinomial to be factored by grouping.

Step 5: Factor by Grouping

Group the terms and factor each group separately: \[ (t^2 + 12t) + (-2t - 24) \]. Factor out the greatest common factor from each group: \[ t(t + 12) - 2(t + 12) \].

Step 6: Factor Out the Common Binomial

Notice that \((t + 12)\) is common in both terms. Factor it out: \[ (t + 12)(t - 2) \]. Hence, the completely factored form is \((t + 12)(t - 2)\).

Key Concepts

Polynomial Factoring

Polynomial factoring is a fundamental skill in algebra that involves breaking down a polynomial into products of simpler polynomials. This is similar to how one might factor numbers in arithmetic, like factoring 12 into 3 and 4. Factoring polynomials is crucial because it simplifies equations and makes them easier to solve or analyze.

Factoring is used to:

• Simplify expressions by breaking them into smaller, more manageable parts.
• Find the roots or solutions of polynomial equations by setting each factor equal to zero.
• Solve word problems involving quadratic equations in various fields like physics or economics.

To effectively factor a polynomial, you generally start by examining the expression to determine if there are any common factors among the terms, and then utilize strategies like the AC method or grouping to further simplify.

AC Method

The AC method is a popular technique for factoring quadratic trinomials, especially when the leading coefficient is not 1. It breaks down the process into clear steps, making it more manageable.

To use the AC method, follow these steps:

• Identify the coefficients: For the trinomial in standard form, like \( ax^2 + bx + c \), identify \( a \), \( b \), and \( c \).
• Calculate \( a \times c \), the product of the leading coefficient and the constant term.
• Find two numbers that multiply to \( a \times c \) and add up to \( b \). These will be your guide to splitting the middle term.
• Rewrite the middle term using these numbers, and then proceed to factor by grouping.

This approach transforms a challenging trinomial into a simpler factoring problem, making it more approachable and less prone to errors.

Factor by Grouping

Factor by grouping is a technique used to simplify polynomial expressions, particularly when dealing with four terms. It involves arranging terms into groups and factoring out the common factors from each group.

Here’s how to factor by grouping:

• Divide the polynomial into two groups. This is typically done after rewriting the expression, such as breaking the middle term as shown in the AC method.
• In each group, factor out the greatest common factor (GCF).
• If done correctly, the two groups will share a common binomial factor, which can then be factored out.

In our example, we rewrote the trinomial into groups, then factored \( t(t + 12) \) and \(-2(t + 12) \), enabling us to factor out \((t + 12)\) from both.

Quadratic Equations

Quadratic equations are a type of polynomial equation with the form \( ax^2 + bx + c = 0 \). These equations can model various scenarios in mathematics and applied sciences, ranging from projectile motion to optimization problems.

Key features of quadratic equations include:

• They can have 0, 1, or 2 real roots, which are the solutions where the equation equals zero.
• The graph of a quadratic equation is a parabola, which opens upwards if \( a \) is positive and downwards if \( a \) is negative.
• Solving quadratics often involves techniques like factoring, completing the square, or using the quadratic formula.

Learning to factor quadratics, as we did in the example, gives you a powerful tool to find solutions efficiently. Factoring reveals the equation's roots by expressing it as a product of binomials, which are then each set to zero to solve for the variable.