Problem 59 - Algebra A Combined Function

Question

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1). Don't forget to factor out the GCF first. $$ 2 t^{5}-14 t^{4}+24 t^{3} $$

Step-by-Step Solution

Answer

The complete factorization is \( 2t^3(t - 3)(t - 4) \).

Step 1: Identify the Greatest Common Factor (GCF)

The given trinomial is \( 2t^5 - 14t^4 + 24t^3 \). First, identify the greatest common factor of all the terms. All terms have at least \( t^3 \) in common, and numerically, the GCF is 2. Thus, the GCF of the entire expression is \( 2t^3 \).

Step 2: Factor Out the GCF

Factor the GCF, \( 2t^3 \), out of each term in the expression. This gives us:\[ 2t^3(t^2 - 7t + 12) \].

Step 3: Factor the Quadratic Trinomial

Now, focus on the quadratic trinomial \( t^2 - 7t + 12 \). Look for two numbers that multiply to 12 (the constant term) and add to -7 (the coefficient of the linear term). The numbers -3 and -4 satisfy these conditions, so the trinomial factors as:\[(t - 3)(t - 4)\].

Step 4: Write the Complete Factorization

Combine the GCF factored out earlier with the newly factored trinomial to obtain the complete factorization of the original expression:\[ 2t^3(t - 3)(t - 4) \].

Key Concepts

Greatest Common Factor (GCF)

In polynomial factorization, identifying the Greatest Common Factor (GCF) is often the first step. The GCF is the largest factor shared by all terms in the expression.
To find the GCF, start by identifying the factors of both the coefficients and the variables in the polynomial.

• Look at the coefficients: For each term of the polynomial, identify any common numerical factor. In the trinomial from our original exercise, the coefficients are 2, -14, and 24. The GCF of these numbers is 2.
• Look at the variable factors: Identify the smallest power of common variables across all terms. Here, each term has at least a factor of \( t^3 \).

With this, the GCF of \( 2t^5 - 14t^4 + 24t^3 \) is \( 2t^3 \). Once the GCF is determined, factor it out from every term to simplify the polynomial, taking us to the next steps in factorization.

Quadratic Trinomial

A quadratic trinomial is a polynomial with three terms and a degree of 2. It commonly takes the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Factoring it requires finding two numbers that:

• Multiply to \( c \) (the constant term).
• Add to \( b \) (the linear coefficient).

For example, let's consider the quadratic trinomial \( t^2 - 7t + 12 \). We need to identify two numbers which multiply to 12 (the constant term) and add to -7 (the coefficient of \( t \)). These numbers are -3 and -4:

• \((-3) \times (-4) = 12\)
• \((-3) + (-4) = -7\)

Thus, \( t^2 - 7t + 12 \) can be factored as \((t - 3)(t - 4)\). This step simplifies the polynomial and prepares it for constructing the complete factorization.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors. It's similar to "undoing" the distributive property. For the original problem \( 2t^5 - 14t^4 + 24t^3 \), factorization is achieved by systematically breaking down the expression into simpler components.Start with these steps:

• First, identify and factor out the Greatest Common Factor (\( 2t^3 \) in this case), simplifying the expression to \( 2t^3(t^2 - 7t + 12) \).
• Then factor the remaining quadratic trinomial, \( t^2 - 7t + 12 \), to obtain \((t - 3)(t - 4)\).

Finally, combine all the factors to express the original polynomial entirely in its factored form: \( 2t^3(t - 3)(t - 4) \). This complete factorization provides a simpler representation and reveals potential roots or solutions of the polynomial equation.