Problem 47 - Elementary and Intermediate Algebra

Question

Factor by grouping. $$ t^{3}-3 t^{2}-7 t+21 $$

Step-by-Step Solution

Answer

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

Step 1: Group the Terms

The first step in factoring by grouping is to divide the polynomial into two groups. This will help us to identify common factors within these groups. Take the polynomial: \[ t^3 - 3t^2 - 7t + 21 \]Group it into:\[ (t^3 - 3t^2) ext{ and } (-7t + 21) \]

Step 2: Factor Out the Greatest Common Factor in Each Group

Now, we factor out the greatest common factor from each group.For the first group \( t^3 - 3t^2 \):- The common factor is \( t^2 \), so after factoring we have \( t^2(t - 3) \).For the second group \( -7t + 21 \):- The common factor is \( -7 \), so after factoring we have \( -7(t - 3) \).

Step 3: Factor Out the Common Binomial Factor

Now that both groups have a common binomial factor of \( t - 3 \), we factor this out:\[ t^2(t - 3) - 7(t - 3) = (t^2 - 7)(t - 3) \]

Step 4: Combine the Factors

The expression is now in its factored form, which combines the common factor from the previous step:\[ (t^2 - 7)(t - 3) \]Verify by multiplying to ensure it matches the original polynomial. The exercise is complete!

Key Concepts

Polynomial Factoring

Polynomial factoring is a process used to express a polynomial as a product of simpler polynomials. It's like breaking down complex numbers into their prime components, which helps us simplify expressions and solve equations. In this exercise, we started with the polynomial \( t^3 - 3t^2 - 7t + 21 \).
To factor, we grouped the polynomial into two parts: \( (t^3 - 3t^2) \) and \( (-7t + 21) \). Breaking it up makes it easier to find common factors in each section. By the end of the problem, the factorization allowed us to rewrite the expression as \( (t^2 - 7)(t - 3) \).
This technique not only simplifies the polynomial but also makes it easier to solve for the variable when set to zero, aiding in finding the roots of the equation. Factoring by grouping is especially useful for cubics and higher-degree polynomials.
Use this technique frequently to enhance your abilities in algebra and beyond!

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides two or more numbers. When factoring by grouping, identifying the GCF is crucial as it assists in simplifying each term of the polynomial seamlessly.
In our exercise, we identified the GCF for each group of terms. For the first set, \( t^3 - 3t^2 \), the GCF was \( t^2 \). Factoring out, we obtained \( t^2(t - 3) \).
For the second set, \( -7t + 21 \), the GCF was \(-7\). After factoring out, it simplified to \( -7(t - 3) \).

• To find the GCF in a term, list out the factors of each term.
• Identify the highest number that appears in each list.

Finding the GCF is a powerful tool in algebra to reduce expressions, solve equations, and tackle factoring problems efficiently.

Binomial Factorization

Binomial factorization involves expressing a polynomial as the product of two binomials. This is a vital aspect of the grouping strategy used in factoring.
After factoring the individual groups in our exercise, we noticed both resulted in the factor \( (t - 3) \). This common binomial factor allows us to write the expression as \( t^2(t - 3) - 7(t - 3) \), which was then simplified to \( (t^2 - 7)(t - 3) \).
The presence of a common binomial factor is the key to combining the groups effectively into a single expression. This technique is particularly beneficial when dealing with polynomials that are not straightforward to factor as the differences of squares or quadratics.
Understand that recognizing and using binomial factors can greatly simplify solving algebraic equations, especially when working with higher-degree polynomials.