Factor $x^3-27$
A) $(x-3)(x^2-3x+9)$
B) $(x+3)(x^2+3x+9)$
C) $(x+3)(x^2-3x+9)$
D) $(x-3)(x^2+3x+9)$
Answer (2)
Recognize the expression as a difference of cubes: x 3 − 27 = x 3 − 3 3 .
Apply the difference of cubes factorization formula: a 3 − b 3 = ( a − b ) ( a 2 + ab + b 2 ) with a = x and b = 3 .
Substitute and simplify: ( x − 3 ) ( x 2 + 3 x + 9 ) .
The factored form is ( x − 3 ) ( x 2 + 3 x + 9 ) .
Explanation
Recognizing the Difference of Cubes We are asked to factor the expression x 3 − 27 . This expression is a difference of cubes, which can be factored using a specific formula.
Applying the Formula The difference of cubes factorization formula is given by: a 3 − b 3 = ( a − b ) ( a 2 + ab + b 2 ) In our case, we have x 3 − 27 , which can be written as x 3 − 3 3 . Thus, a = x and b = 3 .
Substituting and Simplifying Substituting a = x and b = 3 into the formula, we get: x 3 − 3 3 = ( x − 3 ) ( x 2 + x ⋅ 3 + 3 2 ) Simplifying this expression, we have: ( x − 3 ) ( x 2 + 3 x + 9 )
Final Answer Therefore, the factored form of x 3 − 27 is ( x − 3 ) ( x 2 + 3 x + 9 ) . Comparing this with the given options, we see that it matches option D.
Examples
Factoring polynomials like x 3 − 27 is useful in many areas of mathematics and engineering. For example, when designing a bridge, engineers need to analyze the forces acting on the structure. These forces can often be modeled using polynomial equations, and factoring these equations can help engineers determine the points of maximum stress and strain. Similarly, in signal processing, factoring polynomials is used to design filters that remove unwanted noise from signals. By factoring the polynomial that represents the filter, engineers can identify the frequencies that the filter will block or allow to pass through.
The expression x 3 − 27 factors to ( x − 3 ) ( x 2 + 3 x + 9 ) using the difference of cubes formula. Thus, the correct choice is option A: ( x − 3 ) ( x 2 + 3 x + 9 ) .
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