Problem 80 - Elementary and Intermediate Algebra

Question

Factor. $$ 32 b^{6}+80 b^{5} c+50 b^{4} c^{2} $$

Step-by-Step Solution

Answer

The expression factors to \(2b^4(4b + 5c)^2\).

Step 1: Identify the Greatest Common Factor

First, look for the greatest common factor (GCF) in the expression. Here we have 32, 80, and 50 as coefficients and the smallest power of b is 4. The GCF for the coefficients is 2, and the smallest power of b is 4 (i.e., \(b^4\)). Therefore, the GCF of the entire expression is \(2b^4\).

Step 2: Factor Out the GCF

Divide each term in the expression by the GCF (\(2b^4\)) and factor it out. - \(32b^6 \div 2b^4 = 16b^2\)- \(80b^5c \div 2b^4 = 40bc\)- \(50b^4c^2 \div 2b^4 = 25c^2\)This gives us: \(2b^4(16b^2 + 40bc + 25c^2)\).

Step 3: Recognize the Quadratic Form

The resulting expression inside the parentheses, \(16b^2 + 40bc + 25c^2\), is a quadratic trinomial. Notice that it resembles a perfect square trinomial.

Step 4: Identify a Perfect Square Trinomial

Evaluate whether \(16b^2 + 40bc + 25c^2\) is a perfect square trinomial. The first term is \((4b)^2\), the last term is \((5c)^2\), and the middle term can be written as \(2 \times 4b \times 5c = 40bc\). This confirms that it is a perfect square trinomial.

Step 5: Express the Trinomial as a Square

Since \(16b^2 + 40bc + 25c^2\) is a perfect square trinomial, it can be expressed as the square of a binomial: \((4b + 5c)^2\).

Step 6: Write the Final Factored Form

Combine the GCF and the square of the binomial to write the completely factored expression: \(2b^4(4b + 5c)^2\).

Key Concepts

Greatest Common Factor

In mathematics, especially when dealing with polynomials, finding the greatest common factor (GCF) is an essential first step in simplifying or factoring expressions. The GCF is the largest factor that divides each term of the polynomial without leaving a remainder.

To find the GCF:

• Identify the coefficients of each term in the polynomial. For example, in the expression \(32b^6 + 80b^5c + 50b^4c^2\), these coefficients are 32, 80, and 50.
• Determine the greatest number that can divide all coefficients. In this case, it's 2.
• Identify the common variable(s) in each term and determine the smallest power among them. Here, the smallest power of \(b\) is \(b^4\).

By combining these steps, you identify that the GCF of the polynomial is \(2b^4\). Factoring out the GCF simplifies the polynomial and is an important precursor to further factorizations.

Quadratic Trinomial

A quadratic trinomial is a polynomial with three terms, and it's in the form \(ax^2 + bx + c\). In the expression obtained after factoring out the GCF from \(32b^6 + 80b^5c + 50b^4c^2\), we have \(16b^2 + 40bc + 25c^2\).

This matches the quadratic trinomial format where:

• \(a = 16\), which is the coefficient of \(b^2\).
• \(b = 40\), which is the coefficient of \(bc\).
• \(c = 25\), which is the constant term \(c^2\).

Quadratic trinomials can often be factored further. Recognizing this form helps you decide the appropriate method to factor it, such as checking whether it forms a perfect square trinomial.

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic trinomial that can be expressed as the square of a binomial. Recognizing this form allows us to factor the polynomial quickly.

For a trinomial \(a^2 + 2ab + b^2\), the corresponding binomial is \((a + b)^2\). Consider the expression \(16b^2 + 40bc + 25c^2\):

• \(16b^2\) is \((4b)^2\).
• \(25c^2\) is \((5c)^2\).
• The middle term, \(40bc\), is \(2 \times 4b \times 5c\).

This confirms that the expression is a perfect square trinomial because it follows the form \((a + b)^2\) where \(a = 4b\) and \(b = 5c\). Factoring it gives \((4b + 5c)^2\). This step substantially simplifies the polynomial into a product involving repetition of a simple expression.