Which of the following is not a factor of [tex]$x^3-3 x^2-4 x+12$[/tex]?
A) [tex]$(x+3)$[/tex]
B) [tex]$(x-3)$[/tex]
C) [tex]$(x-2)$[/tex]
D) [tex]$(x+2)$[/tex]
Answer (2)
Evaluate the polynomial p ( x ) = x 3 − 3 x 2 − 4 x + 12 at x = − 3 , 3 , 2 , − 2 .
p ( − 3 ) = − 30 , p ( 3 ) = 0 , p ( 2 ) = 0 , p ( − 2 ) = 0 .
Since p ( − 3 ) e q 0 , ( x + 3 ) is not a factor.
Therefore, the answer is ( x + 3 ) .
Explanation
Understanding the Problem We are given the polynomial x 3 − 3 x 2 − 4 x + 12 and asked to determine which of the following is not a factor: ( x + 3 ) , ( x − 3 ) , ( x − 2 ) , or ( x + 2 ) . We can use the factor theorem to check each option. The factor theorem states that if ( x − a ) is a factor of a polynomial p ( x ) , then p ( a ) = 0 .
Setting up the Solution Let p ( x ) = x 3 − 3 x 2 − 4 x + 12 . We will evaluate p ( x ) for each of the given options.
Checking (x+3) First, let's check ( x + 3 ) . This corresponds to x = − 3 . We have p ( − 3 ) = ( − 3 ) 3 − 3 ( − 3 ) 2 − 4 ( − 3 ) + 12 = − 27 − 27 + 12 + 12 = − 30 Since p ( − 3 ) = − 30 e q 0 , ( x + 3 ) is not a factor of the polynomial.
Checking (x-3) Next, let's check ( x − 3 ) . This corresponds to x = 3 . We have p ( 3 ) = ( 3 ) 3 − 3 ( 3 ) 2 − 4 ( 3 ) + 12 = 27 − 27 − 12 + 12 = 0 Since p ( 3 ) = 0 , ( x − 3 ) is a factor of the polynomial.
Checking (x-2) Now, let's check ( x − 2 ) . This corresponds to x = 2 . We have p ( 2 ) = ( 2 ) 3 − 3 ( 2 ) 2 − 4 ( 2 ) + 12 = 8 − 12 − 8 + 12 = 0 Since p ( 2 ) = 0 , ( x − 2 ) is a factor of the polynomial.
Checking (x+2) Finally, let's check ( x + 2 ) . This corresponds to x = − 2 . We have p ( − 2 ) = ( − 2 ) 3 − 3 ( − 2 ) 2 − 4 ( − 2 ) + 12 = − 8 − 12 + 8 + 12 = 0 Since p ( − 2 ) = 0 , ( x + 2 ) is a factor of the polynomial.
Conclusion Since p ( − 3 ) e q 0 , ( x + 3 ) is not a factor of the polynomial. The other options, ( x − 3 ) , ( x − 2 ) , and ( x + 2 ) , are factors of the polynomial.
Examples
Factoring polynomials is a fundamental concept in algebra and has practical applications in various fields. For instance, in engineering, factoring can help simplify complex equations that model physical systems, making them easier to analyze and design. In computer graphics, polynomial factorization can be used to optimize rendering algorithms, improving the efficiency of image generation. Understanding factors and roots of polynomials is also crucial in cryptography, where it forms the basis for secure communication protocols.
After evaluating the polynomial p ( x ) = x 3 − 3 x 2 − 4 x + 12 at specific values, we find that the only option not yielding zero is ( x + 3 ) . Thus, ( x + 3 ) is not a factor of the polynomial. The correct answer is ( x + 3 ) .
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