Answer:
B [tex] ( {x}^{2} - 2)( {x}^{2} + 2)( {x}^{4} + 4)[/tex]
Step-by-step explanation:
[tex] {x}^{8} - 16 \\ = ( {x}^{4} )^{2} - {4}^{2} \\ = ( {x}^{4} - 4) ( {x}^{4} + 4) \\ = [ ({x}^{2})^{2} - (2) ^{2}] ( {x}^{4} + 4) \\ = ( {x}^{2} - 2)( {x}^{2} + 2)( {x}^{4} + 4) \\ [/tex]
Hence, option B is the correct answer.
The given polynomial x⁸ − 16 can be factorised as (x² − 2)(x² + 2)(x⁴ + 4). The correct option is B.
Polynomials are mathematical expressions involving variables raised with non-negative integers and coefficients(constants who are in multiplication with those variables) and constants with only operations of addition, subtraction, multiplication and non-negative exponentiation of variables involved.
One of the most used algebraic identities is a²-b². It is expanded as,
a² - b² = (a+b)(a-b)
The given polynomial can be factorised using the algebraic identity shown above as,
x⁸ − 16
= (x⁴)² − (4)²
Using the identity (x⁴)² − (4)²,
= (x⁴ − 4)(x⁴ + 4)
= [(x²)² − (2)²] (x⁴ + 4)
Using the identity for (x²)² − (2)²,
= (x² − 2)(x² + 2)(x⁴ + 4)
Hence, the given polynomial x⁸ − 16 can be factorised as (x² − 2)(x² + 2)(x⁴ + 4).
Learn more about polynomials here:
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