Problem 36 - Elementary and Intermediate Algebra

Question

Factor each polynomial by factoring out the opposite of the \(G C F .\) $$ -21 t^{5}+28 t^{3} $$

Step-by-Step Solution

Answer

The factored form is \(-7t^3 (3t^2 - 4)\).

Step 1: Identify the Greatest Common Factor (GCF)

Examine the coefficients and the variables in the terms \(-21t^5\) and \(+28t^3\). The numerical GCF of both \(-21\) and \(28\) is \(7\). The GCF of \(t^5\) and \(t^3\) is \(t^3\), since both terms have at least three \(t\) factors. Thus, the GCF of the polynomial is \(7t^3\).

Step 2: Factor out the GCF

Since we're asked to factor out the opposite of the GCF, we need to factor out \(-7t^3\) instead of \(7t^3\). Divide each term of the polynomial by \(-7t^3\). For \(-21t^5\), the result is \(3t^2\) because \(\frac{-21t^5}{-7t^3} = 3t^2\). For \(+28t^3\), the result is \(-4\), because \(\frac{28t^3}{-7t^3} = -4\).

Step 3: Write the factored expression

After factoring out \(-7t^3\), the expression is transformed into:\(-7t^3(3t^2 - 4)\). This is the polynomial expressed in its factored form.

Key Concepts

Greatest Common Factor (GCF)

The concept of the Greatest Common Factor (GCF) is pivotal in factoring polynomials. Understanding the GCF means identifying the largest factor that divides each term in an algebraic expression.
When you look at terms like

• \(-21t^5\)
• \(+28t^3\)

you need to dissect both the numerical and variable parts independently. The numerical part of the GCF, in this case, is determined by finding the greatest number that can divide both \(-21\) and \(28\). Here, that number is \(7\).
Regarding the variables, each term with a \(t\) has an exponent; the GCF for \(t^5\) and \(t^3\) is the lowest power shared, which is \(t^3\). These combined give us the GCF, \(7t^3\). Recognizing the GCF is crucial to simplifying expressions efficiently.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations, which collectively form terms. Each term in an algebraic expression can include both numerical coefficients and variable components. In our example, the expression \(-21t^5 + 28t^3\) comprises two distinct terms:

• \(-21t^5\)
• \(+28t^3\)

Each term contains a numerical coefficient (

• \(-21\)
• \(+28\)

) paired with a power of \(t\), the variable.
Understanding the parts of algebraic expressions helps pinpoint common factors among terms, laying the groundwork for successful factoring. Each part of the expression speaks to the operations and relationships the expression conveys, serving as building blocks in more complex algebraic operations.

Factoring Techniques

Factoring is the process of rewriting a polynomial as a product of its factors, making expressions simpler to solve or evaluate. There are several techniques, but a primary and foundational approach is factoring out the GCF. In the task at hand, an additional twist involves factoring out the opposite of the GCF.
Here's how we apply this technique. We already identified \(7t^3\) as the GCF, but since we're factoring out the *opposite*, we use \(-7t^3\). Dividing each polynomial term by \(-7t^3\) leads to:

• For \(-21t^5\), dividing yields \((-21t^5)/(-7t^3) = 3t^2\).
• For \(+28t^3\), dividing yields \((28t^3)/(-7t^3) = -4\).

These results transform the expression into a factored form: \(-7t^3(3t^2 - 4)\).
Mastering techniques like these allows you to manipulate and simplify algebraic expressions effectively, making them more manageable for any further mathematical operations.