Problem 69 - Beginning and Intermediate Algebra

Question

Factor completely, if possible. Begin by asking yourself, "Can I factor out a GCF?" $$2 r^{4}+26 r^{3}+84 r^{2}$$

Step-by-Step Solution

Answer

The completely factored expression is: \(2r^2 (r+6)(r+7)\).

Step 1: Identify the GCF

In order to find the GCF of the terms, we need to look at the coefficients of each term and the degree of each term. Coefficients: 2, 26, 84 Degree of r: 4, 3, 2 Find the GCF of the coefficients and the lowest degree of r.

Step 2: Find the GCF of the coefficients

The coefficients are 2, 26, and 84. The GCF of these numbers is 2.

Step 3: Find the lowest degree of r

The degrees of r in the terms are 4, 3, and 2. The lowest degree of r is 2.

Step 4: Factor out the GCF

Now, we'll factor out the GCF we found from the polynomial. GCF: 2r^2 So our expression becomes: \(2r^2 (r^2 + 13r + 42)\)

Step 5: Factor the quadratic inside the parenthesis

Now we need to factor the quadratic inside the parenthesis: \(r^2 + 13r + 42\). To factor it, we need to find two numbers whose product is equal to the last term (42) and whose sum is equal to the middle term (13). These two numbers are 6 and 7. Therefore, we can factor the quadratic as: \((r+6)(r+7)\)

Step 6: Write the complete factored expression

Now, we can write the completely factored expression, combining the GCF and the factored quadratic: \[2r^2 (r+6)(r+7)\] So, the completely factored expression is: \[2r^2 (r+6)(r+7)\]

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is essential in simplifying polynomials. It is the largest factor shared by all the terms of the polynomial. To find the GCF, consider both the numerical coefficients and the variables with their exponents.
In the given polynomial, we examine the coefficients: 2, 26, and 84. We see that the common factor here is 2. This is because 2 is the largest number by which all these coefficients can be divided evenly.
Next, look at the variable part. Each term in the polynomial involves the variable \(r\) with different powers: 4, 3, and 2. The smallest power of \(r\) among these is 2, which means \(r^2\) is the common factor for the variable part across all terms.
Combining these, the GCF for the polynomial \(2r^4 + 26r^3 + 84r^2\) is \(2r^2\). Factoring out the GCF simplifies the polynomial significantly and is the first step before dealing with any remaining quadratic expression.

Quadratic Factoring

Once the GCF has been factored out, we are left with a quadratic expression: \(r^2 + 13r + 42\). We need to factor this quadratic further to completely simplify the polynomial.
Quadratic factoring involves finding two numbers that multiply to give the constant term (in this case, 42) and add up to the middle coefficient (13).

• Product needed: 42
• Sum needed: 13

To find these numbers, examine the pairs that multiply to 42. Among these pairs, 6 and 7 satisfy both criteria: they add up to 13 and multiply to 42.
So, the expression \(r^2 + 13r + 42\) factors into a product of two binomials: \((r + 6)(r + 7)\). This method of splitting involves trial and error with the pairs, but it becomes intuitive with practice.

Polynomial Degrees

Understanding polynomial degrees is fundamental when working with polynomials, especially during factoring.
The degree of a term is determined by the sum of the exponents of the variables in that term. In a polynomial, the degree is the highest degree among the terms.
For the polynomial given, \(2r^4 + 26r^3 + 84r^2\), observe the degrees:

• First term: degree 4 (from \(r^4\))
• Second term: degree 3 (from \(r^3\))
• Third term: degree 2 (from \(r^2\))

Here, the overall degree of the polynomial is 4, corresponding to the term with the highest power of \(r\).
Knowing the degrees helps in identifying the GCF since the smallest degree provides the lowest power of \(r\) that should be factored out. This makes polynomial operations and simplifications easier and helps maintain clarity throughout the factoring process.