Factor completely.
$3 w^6-32 w^5+20 w^4$
To factor the expression \(3w^6 - 32w^5 + 20w^4\), we first look for the greatest common factor (GCF) of the three terms. Each term contains at least \(w^4\), and there are no numerical common factors among 3, -32, and 20. Thus, we can factor out \(w^4\):
\[3w^6 - 32w^5 + 20w^4 = w^4(3w^2 - 32w + 20).\]
Now, we need to factor the quadratic trinomial \(3w^2 - 32w + 20\). To do this, we look for two numbers that multiply to \(3 \times 20 = 60\) and add up to \(-32\). We can test factor pairs of 60:
Since -2 and -30 sum to -32, we can split the middle term:
\[3w^2 - 2w - 30w + 20.\]
Next, we group the terms to factor the expression by grouping:
\[(3w^2 - 2w) - (30w - 20)\]
\[w(3w - 2) - 10(3w - 2)\]
Now, factor out the common binomial factor \((3w - 2)\):
\[(w - 10)(3w - 2).\]
Combining this with the factor \(w^4\) we extracted earlier, the fully factored form of the expression is \(w^4(w - 10)(3w - 2)\).
Factoring complex expressions is a foundational skill for solving polynomial equations in fields like physics and engineering. For example, if you are modeling the trajectory of an object where the position is given by a high-degree polynomial like \(3t^6 - 32t^5 + 20t^4\), factoring allows you to easily identify the times \(t\) when the object is at a specific position (like the ground, where position is zero). In this case, solving \(w^4(w - 10)(3w - 2) = 0\) reveals the roots \(w = 0\), \(w = 10\), and \(w = \frac{2}{3}\), which are critical points in the analysis of the motion.