Problem 75 - Beginning and Intermediate Algebra

Question

Factor completely, if possible. Begin by asking yourself, "Can I factor out a GCF?" $$(x+y) t^{2}-4(x+y) t-21(x+y)$$

Step-by-Step Solution

Answer

The completely factored expression for the given polynomial is \((x+y)(t - 7)(t + 3)\).

Step 1: Identify the GCF of the terms in the polynomial

We can observe that all the terms in the polynomial have (x+y) in common. So, the GCF is (x+y).

Step 2: Factor out the GCF

Now we factor out the GCF from each term of the polynomial: \((x+y)t^2 - 4(x+y)t - 21(x+y) = (x+y)(t^2 - 4t - 21)\)

Step 3: Factor the quadratic polynomial inside the parentheses

Next, we need to factor the quadratic polynomial \(t^2 - 4t - 21\). We are looking for two numbers that multiply to get -21 and add up to -4. Upon trying different combinations, we realize that (-7) and (3) fit this criterion since (-7) x (3) = -21 and (-7) + (3) = -4. Therefore, we can now factor the quadratic polynomial. \(t^2 - 4t - 21 = (t - 7)(t + 3)\)

Step 4: Write the completely factored expression

Finally, by substituting the factored quadratic polynomial back into the expression, we have: \((x+y)(t^2 - 4t - 21) = (x+y)((t - 7)(t + 3))\) Thus, the completely factored expression for the given polynomial is: \((x+y)(t - 7)(t + 3)\)

Key Concepts

Greatest Common Factor (GCF)

Finding the Greatest Common Factor (GCF) is often the first step in factoring polynomials. The GCF is the largest expression that can be evenly divided out of each term within a polynomial.
To determine the GCF, follow these steps:

• Examine each term in the polynomial to find common elements.
• The GCF will be the product of these common elements, which can be numbers, variables, or both.

In the exercise, the common factor across every term is \(x+y\).
Factoring out \(x+y\) simplifies the expression and allows further factoring.

Quadratic Polynomial

A quadratic polynomial is an expression of the form \(ax^2 + bx + c\). It is called "quadratic" because the highest degree of the variable is 2.
Factoring quadratics is a key algebra skill and often involves finding two numbers that multiply to \(ac\) and add to \(b\).
For \(t^2 - 4t - 21\), it is crucial to:

• Identify the coefficients: here, \(a=1\), \(b=-4\), and \(c=-21\).
• Multiply \(a\) by \(c\) to get \(-21\) and find two numbers that multiply to this product and add up to \(-4\).

The successful numbers are -7 and 3, which allow you to factor the quadratic into \((t-7)(t+3)\).
This method requires practice but is essential for solving quadratic equations.

Factoring Techniques

Factoring techniques are diverse and choosing the right strategy depends on the polynomial at hand.
Here are some common methods:

• Factoring out the GCF, as shown earlier, to simplify expressions.
• Choosing the correct strategy for quadratics, like finding two numbers that multiply and add up to specific coefficients.
• Using trial and error or recognizing patterns like the difference of squares.

Completely factoring a polynomial involves observing its structure and systematically applying these techniques.
In our case, after factoring out \(x+y\), we used the technique of factoring a quadratic by decomposing middle terms.
This highlights the importance of being familiar with all methods, making the process of factoring smoother and more efficient.