Which polynomial is factored completely?
A. [tex]$121 x^2+36 y^2$[/tex]
B. [tex]$(4 x+4)(x+1)$[/tex]
C. [tex]$2 x(x^2-4)$[/tex]
D. [tex]$3 x^4-15 n^3+12 n^2$[/tex]
Answer (2)
Check each polynomial to see if it can be factored further.
121 x 2 + 36 y 2 is a sum of squares and cannot be factored further.
( 4 x + 4 ) ( x + 1 ) can be factored as 4 ( x + 1 ) 2 .
2 x ( x 2 − 4 ) can be factored as 2 x ( x − 2 ) ( x + 2 ) .
3 x 4 − 15 x 3 + 12 x 2 can be factored as 3 x 2 ( x − 1 ) ( x − 4 ) .
The polynomial that is factored completely is 121 x 2 + 36 y 2 .
Explanation
Problem Analysis We are given four polynomials and asked to identify the one that is factored completely. A polynomial is factored completely if it cannot be factored any further. We will examine each option.
Checking Each Polynomial
121 x 2 + 36 y 2 : This is a sum of squares, ( 11 x ) 2 + ( 6 y ) 2 . A sum of squares cannot be factored further using real numbers. Therefore, this polynomial is factored completely.
( 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, this polynomial was not factored completely.
2 x ( x 2 − 4 ) : The term x 2 − 4 is a difference of squares, which can be factored as ( x − 2 ) ( x + 2 ) . Thus, the polynomial becomes 2 x ( x − 2 ) ( x + 2 ) . Since it can be factored further, this polynomial was not factored completely.
3 x 4 − 15 x 3 + 12 x 2 : We can factor out 3 x 2 from each term: 3 x 2 ( x 2 − 5 x + 4 ) . The quadratic x 2 − 5 x + 4 can be factored as ( x − 1 ) ( x − 4 ) . Thus, the polynomial becomes 3 x 2 ( x − 1 ) ( x − 4 ) . Since it can be factored further, this polynomial was not factored completely.
Conclusion Therefore, the polynomial that is factored completely is 121 x 2 + 36 y 2 .
Examples
Factoring polynomials is a fundamental concept in algebra and has many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or circuits. Architects use factoring to optimize space and materials when designing buildings. In cryptography, factoring large numbers is crucial for securing data transmissions. Understanding how to factor polynomials completely allows us to solve problems more efficiently and accurately in various fields.
The polynomial that is factored completely is 121 x 2 + 36 y 2 as it is a sum of squares and cannot be factored further. Other options can be factored further. Therefore, Option A is the correct choice.
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