The polynomial x3 + 8 is equal to
The Correct Answer and Explanation is:
The polynomial ( x^3 + 8 ) can be factored using the sum of cubes formula. The sum of cubes formula is:
[
a^3 + b^3 = (a + b)(a^2 – ab + b^2)
]
In the given polynomial ( x^3 + 8 ), we can recognize this as a sum of cubes, since ( 8 ) is equal to ( 2^3 ). Therefore, we can rewrite the polynomial as:
[
x^3 + 8 = x^3 + 2^3
]
Now, using the sum of cubes formula, where ( a = x ) and ( b = 2 ), we can factor the polynomial as:
[
x^3 + 2^3 = (x + 2)(x^2 – 2x + 2^2)
]
Simplifying the second factor:
[
x^2 – 2x + 4
]
Thus, the factored form of ( x^3 + 8 ) is:
[
x^3 + 8 = (x + 2)(x^2 – 2x + 4)
]
Explanation:
The sum of cubes formula helps in factoring polynomials of the form ( a^3 + b^3 ). This formula is based on the idea that any number that is the sum of two cubes can be expressed as the product of a binomial and a trinomial. The binomial ( (a + b) ) represents the sum of the cube roots, while the trinomial ( a^2 – ab + b^2 ) accounts for the other terms that arise when expanding the product.
In the case of ( x^3 + 8 ), we recognize that ( 8 ) is a perfect cube, ( 2^3 ), and apply the sum of cubes formula with ( a = x ) and ( b = 2 ). This allows us to factor the expression into the product of two terms: ( (x + 2) ) and ( (x^2 – 2x + 4) ).
This factoring method is very useful because it simplifies polynomials and makes them easier to solve or manipulate in algebraic problems. The factored form ( (x + 2)(x^2 – 2x + 4) ) can be used for further analysis, such as finding the roots or solving equations.