Factor completely.
$7 u^7+28 u^6+28 u^5$
We are tasked with factoring the polynomial expression \(7u^7 + 28u^6 + 28u^5\) completely. To begin, we identify the terms in the expression, which are \(7u^7\), \(28u^6\), and \(28u^5\). Our primary objective is to extract the greatest common factor (GCF) from these three terms. The numerical coefficients are 7, 28, and 28. The GCF of 7, 28, and 28 is 7. For the variable part, we look at the powers of \(u\): \(u^7\), \(u^6\), and \(u^5\). The smallest power of \(u\) present is \(u^5\). Therefore, the GCF of the entire expression is \(7u^5\).
Now we factor out the GCF, \(7u^5\), from each term of the expression:
\[7u^7 + 28u^6 + 28u^5 = 7u^5(u^2) + 7u^5(4u) + 7u^5(4)\]
Factoring out \(7u^5\), we get:
\[7u^5(u^2 + 4u + 4)\]
Next, we examine the quadratic expression inside the parentheses, which is \(u^2 + 4u + 4\). This is a perfect square trinomial. We can check if it follows the pattern \(a^2 + 2ab + b^2 = (a+b)^2\), where \(a = u\) and \(b = 2\). Indeed, \(2ab = 2(u)(2) = 4u\) and \(b^2 = 2^2 = 4\). Therefore, \(u^2 + 4u + 4 = (u + 2)^2\).
By combining the GCF we factored out in the previous step and the result of the perfect square trinomial, we arrive at the completely factored form:
\[7u^5(u + 2)^2\]
This expression cannot be factored further over the rational numbers. Thus, the final factored form of the original expression is \(7u^5(u + 2)^2\).
Factoring polynomials is a fundamental tool in algebra used for simplifying complex expressions and solving equations. For example, if you are modeling the growth of a population using an expression like \(7u^7 + 28u^6 + 28u^5\) where \(u\) represents time, factoring it into \(7u^5(u + 2)^2\) allows you to easily find the time values (roots) where the population is zero or identify critical points in the model's behavior. This technique helps bridge the gap between abstract algebraic manipulation and practical problem solving in fields such as engineering, economics, and physics.