factorizations is correct x³ - 64 = (x - 4) (x² + 4x + 16) correct option is b.
Analyze the given expressions
Before attempting to factorize, it's crucial to examine the given expressions.
Expression (a): x + 27 = (x - 4) (x ≥ - 16)
This expression involves a linear term (x + 27) and a constraint (x ≥ - 16).
Expression (b): x³ - 64 = (x - 4) (x² + 4x + 16)
This expression involves a cubic term (x³) and a quadratic term (x² + 4x + 16).
Expression (c): x³ + 64 = (x - 4) (x² + 4x - 16)
Similar to expression (b), this expression involves a cubic term (x³) and a quadratic term (x² + 4x - 16).
Expression (d): x³ - 64 = (x - 4) (x² - 4x + 16)
Again, this expression involves a cubic term (x³) and a quadratic term (x² - 4x + 16).
Factorize the cubic terms
Since the goal is to factorize the given expressions, let's focus on the cubic terms (x³).
Expression (a): x + 27
This expression is not a cubic term. It's a linear term.
Expression (b): x³ - 64
This expression can be factored using the difference of cubes pattern:
(a³ - b³) = (a - b)(a² + ab + b²)
In this case, a = x and b = 4:
(x³ - 4³) = (x - 4)(x² + 4x + 16)
Expression (c): x³ + 64
This expression cannot be factored using the difference of cubes pattern as it involves adding, not subtracting, cubes.
Expression (d): x³ - 64
Similar to expression (b), this expression can be factored using the difference of cubes pattern:
(x³ - 4³) = (x - 4)(x² + 4x + 16)
Compare the factorizations
After factorizing the cubic terms, compare the resulting factorizations:
Expression (a): x + 27 ≠ (x - 4) (x² + 4x + 16) [Incorrect]
Expression (b): x³ - 64 = (x - 4)(x² + 4x + 16) [Correct]
Expression (c): x³ + 64 ≠ (x - 4)(x² + 4x - 16) [Incorrect]
Expression (d): x³ - 64 = (x - 4)(x² + 4x + 16) [Correct]
Therefore, the only correct factorization among the given options is:
Expression (b): x³ - 64 = (x - 4)(x² + 4x + 16)
Which of the following factorizations is correct?
(a) x + 27 = (x - 4) (x≥ - 16)
(b) x³ - 64 = (x - 4) (x² + 4x + 16)
(c) x³ + 64 = (x - 4) (x² + 4x - 16)
(d) x³ - 64 = (x - 4) (x² - 4x + 16)