Factor completely.
$-y^5+6 y^3$
The expression to factor is \(-y^5 + 6y^3\). Our goal is to factor this polynomial completely over the rational numbers. First, we identify the greatest common factor (GCF) of the terms \(-y^5\) and \(6y^3\). The variables are powers of \(y\), specifically \(y^5\) and \(y^3\). The highest power of \(y\) that divides both is \(y^3\). Thus, we can factor out \(y^3\).
Factoring out \(y^3\) from the expression \(-y^5 + 6y^3\), we get:
\[y^3 \cdot \left(\frac{-y^5}{y^3} + \frac{6y^3}{y^3}\right) = y^3(-y^2 + 6)\]
Alternatively, we can factor out \(-y^3\):
\[-y^3(y^2 - 6)\]
We now examine the polynomial \(y^2 - 6\) to see if it can be factored further over the rational numbers. A polynomial of the form \(y^2 - a^2\) can be factored as \((y - a)(y + a)\). Here, \(a^2 = 6\), which means \(a = \sqrt{6}\). Since \(\sqrt{6}\) is not a rational number, the polynomial \(y^2 - 6\) cannot be factored into polynomials with rational coefficients. Therefore, the expression is completely factored over the rational numbers.
Factoring polynomials like \(-y^5 + 6y^3\) is a foundational skill in algebra, used frequently when simplifying complex rational expressions or solving equations in physics and engineering. For example, if you are modeling the motion of an object where the position is described by a polynomial function, factoring the expression can help you identify the points in time (roots) where the object passes through a specific location, such as the origin, making the physics of the system much easier to understand.