Which product of prime polynomials is equivalent to $3 x^4-81 x$?
A. $3 x(x-3)(x^2-3 x-9)$
B. $3 x(x-3)(x^2+3 x+9)$
C. $3 x(x-3)(x-3)(x+3)$
D. $3 x(x-3)(x+3)(x+3)$
Answer (2)
Factor out the common factor: 3 x 4 − 81 x = 3 x ( x 3 − 27 ) .
Factor the difference of cubes: x 3 − 27 = ( x − 3 ) ( x 2 + 3 x + 9 ) .
Combine the factors: 3 x 4 − 81 x = 3 x ( x − 3 ) ( x 2 + 3 x + 9 ) .
The product of prime polynomials is: 3 x ( x − 3 ) ( x 2 + 3 x + 9 ) .
Explanation
Understanding the Problem We are given the expression 3 x 4 − 81 x and asked to factor it into a product of prime polynomials. This means we want to break it down into factors that cannot be factored further.
Factoring out the common factor First, we can factor out the common factor 3 x from the expression: 3 x 4 − 81 x = 3 x ( x 3 − 27 )
Factoring the difference of cubes Next, we recognize that x 3 − 27 is a difference of cubes. We can factor it using the formula a 3 − b 3 = ( a − b ) ( a 2 + ab + b 2 ) . In this case, a = x and b = 3 , so we have: x 3 − 27 = x 3 − 3 3 = ( x − 3 ) ( x 2 + 3 x + 9 )
Combining the factors Therefore, the original expression can be factored as: 3 x 4 − 81 x = 3 x ( x − 3 ) ( x 2 + 3 x + 9 )
Checking the quadratic factor Now, we need to check if the quadratic x 2 + 3 x + 9 can be factored further. We can find the discriminant using the formula Δ = b 2 − 4 a c , where a = 1 , b = 3 , and c = 9 :
Δ = 3 2 − 4 ( 1 ) ( 9 ) = 9 − 36 = − 27 Since the discriminant is negative, the quadratic has no real roots and cannot be factored further using real numbers. Thus, x 2 + 3 x + 9 is a prime polynomial.
Final Answer Comparing our factored expression 3 x ( x − 3 ) ( x 2 + 3 x + 9 ) with the given options, we find that it matches the second option: 3 x ( x − 3 ) ( x 2 + 3 x + 9 )
Conclusion Therefore, the product of prime polynomials equivalent to 3 x 4 − 81 x is 3 x ( x − 3 ) ( x 2 + 3 x + 9 ) .
Examples
Factoring polynomials is a fundamental skill in algebra and has many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or analyzing circuits. Imagine you're designing a bridge, and you have an equation that describes the load distribution. By factoring the equation, you can identify critical points where the load is highest, ensuring the bridge is strong enough to withstand the stress. Similarly, in computer graphics, factoring can help optimize rendering processes by simplifying the calculations needed to display 3D objects. This makes the rendering faster and more efficient, leading to smoother animations and more realistic visuals.
The product of prime polynomials equivalent to 3 x 4 − 81 x is 3 x ( x − 3 ) ( x 2 + 3 x + 9 ) , which corresponds to option B. First, we factored out the common factor of 3 x and recognized the difference of cubes within the expression. Finally, we confirmed that the quadratic factor is prime and cannot be further factored.
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