Which polynomial is factored completely?
$121 x^2+36 y^2$
$(4 x+4)(x+1)$
$2 x\left(x^2-4\right)$
$3 x^4-15 n^3+12 n^2$
Answer (2)
The polynomial that is factored completely is 121 x 2 + 36 y 2 . The other options can be further factored or simplified, indicating they are not completely factored. Thus, the correct answer is 121 x 2 + 36 y 2 .
;
121 x 2 + 36 y 2 is a sum of squares and is factored completely.
( 4 x + 4 ) ( x + 1 ) can be further factored as 4 ( x + 1 ) 2 , so it's not factored completely.
2 x ( x 2 − 4 ) can be further factored as 2 x ( x − 2 ) ( x + 2 ) , so it's not factored completely.
3 x 4 − 15 x 3 + 12 x 2 can be further factored as 3 x 2 ( x − 1 ) ( x − 4 ) , so it's not factored completely.
Therefore, only 121 x 2 + 36 y 2 is factored completely. 121 x 2 + 36 y 2
Explanation
Problem Analysis We need to determine which of the given polynomials are factored completely. A polynomial is factored completely if it is written as a product of irreducible factors. Let's analyze each polynomial separately.
Analyzing the First Polynomial The first polynomial is 121 x 2 + 36 y 2 . This is a sum of squares, and since there is no common factor, it cannot be factored further using real numbers. Therefore, it is factored completely.
Analyzing the Second Polynomial The second polynomial is ( 4 x + 4 ) ( x + 1 ) . We can factor out a 4 from the first term: 4 ( x + 1 ) ( x + 1 ) = 4 ( x + 1 ) 2 . Since we can factor out a constant, the original polynomial is not factored completely.
Analyzing the Third Polynomial The third polynomial is 2 x ( x 2 − 4 ) . The term x 2 − 4 is a difference of squares and can be factored as ( x − 2 ) ( x + 2 ) . Thus, 2 x ( x 2 − 4 ) = 2 x ( x − 2 ) ( x + 2 ) . Since we can factor the quadratic term, the original polynomial is not factored completely.
Analyzing the Fourth Polynomial The fourth polynomial is 3 x 4 − 15 x 3 + 12 x 2 . We can factor out the common factor 3 x 2 to get 3 x 2 ( x 2 − 5 x + 4 ) . The quadratic x 2 − 5 x + 4 can be factored as ( x − 1 ) ( x − 4 ) . Thus, 3 x 4 − 15 x 3 + 12 x 2 = 3 x 2 ( x − 1 ) ( x − 4 ) . Since we can factor the quadratic term, the original polynomial is not factored completely.
Conclusion In summary:
121 x 2 + 36 y 2 is factored completely.
( 4 x + 4 ) ( x + 1 ) is not factored completely.
2 x ( x 2 − 4 ) is not factored completely.
3 x 4 − 15 x 3 + 12 x 2 is not factored completely.
Examples
Factoring polynomials is a fundamental concept in algebra and is used in various real-world applications. For instance, engineers use factoring to simplify complex equations when designing structures or analyzing systems. In economics, factoring can help in modeling supply and demand curves. Computer graphics also utilizes polynomial factoring to create smooth curves and surfaces. By understanding how to factor polynomials completely, students can develop a strong foundation for advanced mathematical concepts and their practical applications.
