Which polynomial is factored completely?
To determine which polynomial is factored completely, we must analyze each given expression to see if it can be simplified further into smaller polynomial factors. A polynomial is considered factored completely when it is written as a product of irreducible polynomials.
We examine the first expression, \(4(4x^4 - 1)\). The binomial \(4x^4 - 1\) can be identified as a difference of squares, specifically \((2x^2)^2 - 1^2\). Applying the difference of squares formula, \(a^2 - b^2 = (a-b)(a+b)\), we get:
\[4x^4 - 1 = (2x^2 - 1)(2x^2 + 1).\]
The expression can be further factored into \(4(2x^2 - 1)(2x^2 + 1)\). Since it can be decomposed further, it is not completely factored.
Next, we examine the second expression, \(2x(y^3 - 4y^2 + 5y)\). We observe that all terms inside the parentheses share a common factor of \(y\). We can extract \(y\) from the expression:
\[y^3 - 4y^2 + 5y = y(y^2 - 4y + 5).\]
Thus, the expression can be written as \(2xy(y^2 - 4y + 5)\). Since there is a common factor that was not extracted, it is not completely factored.
Now, we look at the third expression, \(3x(9x^2 + 1)\). The factor \((9x^2 + 1)\) is a sum of squares. A sum of two squares, \(a^2 + b^2\), is irreducible over the set of real numbers. Since we have already extracted the greatest common factor \(3x\), this expression cannot be factored any further. Therefore, this polynomial is factored completely.
Finally, we check the fourth expression, \(5x^2 - 17x + 14\). To check if it can be factored, we calculate its discriminant \(D = b^2 - 4ac\):
\[D = (-17)^2 - 4(5)(14) = 289 - 280 = 9.\]
Since the discriminant is a perfect square, the quadratic can be factored. The roots are \(\frac{17 \pm 3}{10}\), giving \(x = 2\) and \(x = \frac{14}{10} = \frac{7}{5}\). Thus, the expression can be written as:
\[(x - 2)(5x - 7).\]
Since it can be expressed as a product of two linear factors, the original expression was not factored.
Based on our analysis, the only expression that cannot be factored further is \(3x(9x^2 + 1)\). Thus, it is the only one factored completely. The final answer is \(\boxed{3x(9x^2 + 1)}\).
In engineering and physics, factoring polynomials is essential for simplifying equations that model physical systems, such as calculating the roots of a motion equation to determine when an object hits the ground. When a system of equations is simplified into its irreducible factors, it becomes much easier to identify critical points, such as equilibrium or intersection points, which are vital for designing stable and efficient structures or predicting the behavior of moving components. Understanding complete factorization ensures that no hidden solutions are missed during the analysis.