Factor completely.
$2 b^6-7 b^5-2 b^4+7 b^3$
The polynomial we wish to factor is \(P(b) = 2b^6 - 7b^5 - 2b^4 + 7b^3\). To begin, we observe that every term in the expression contains at least \(b^3\). Therefore, we can factor out the greatest common factor, \(b^3\), from the entire polynomial:
\[P(b) = b^3(2b^3 - 7b^2 - 2b + 7)\]
Now we focus on the cubic polynomial inside the parentheses: \(Q(b) = 2b^3 - 7b^2 - 2b + 7\). We can attempt to factor this by grouping. We group the terms in pairs:
\[Q(b) = (2b^3 - 2b) + (-7b^2 + 7)\]
Next, we factor out the common terms from each group. From the first group, we can factor out \(2b\), and from the second group, we can factor out \(-7\):
\[Q(b) = 2b(b^2 - 1) - 7(b^2 - 1)\]
We notice that both terms share a common factor of \((b^2 - 1)\). Factoring this out gives us:
\[Q(b) = (2b - 7)(b^2 - 1)\]
The term \((b^2 - 1)\) is a difference of squares, which factors further into \((b - 1)(b + 1)\). Substituting this back into our expression for \(Q(b)\), we get:
\[Q(b) = (2b - 7)(b - 1)(b + 1)\]
Finally, we combine the common factor we pulled out in the first step with the factors we just found to obtain the fully factored form of the original polynomial:
\[P(b) = b^3(2b - 7)(b - 1)(b + 1)\]
This is the complete factorization of the given polynomial.
Factoring polynomials is a fundamental tool in algebra used for simplifying expressions, solving equations, and understanding the behavior of functions. For instance, if you are designing a mechanical part whose volume is defined by the function \(V(b) = 2b^6 - 7b^5 - 2b^4 + 7b^3\), factoring the expression allows you to easily identify the values of \(b\) for which the volume is zero (the roots), which are critical for determining physical dimensions and boundaries in engineering. This process helps simplify complex models, making them easier to compute and analyze.