Answer :
To determine the factored form of the expression \( x^9 - 1000 \), follow these steps:
1. Recognize that \( 1000 = 10^3 \):
[tex]\[ x^9 - 1000 = x^9 - 10^3 \][/tex]
2. See if it fits a known algebraic pattern: Here, we are dealing with the difference of cubes, since \( 10^3 \) is the cube of 10. The difference of cubes formula is:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
3. Rewrite the expression using the difference of cubes formula with \( a = x^3 \) and \( b = 10 \):
[tex]\[ x^9 - 10^3 = (x^3)^3 - 10^3 = (x^3 - 10)\left((x^3)^2 + (x^3)(10) + 10^2\right) \][/tex]
4. Simplify the quadratic term in the factorization:
[tex]\[ (x^3 - 10)\left(x^6 + 10x^3 + 100\right) \][/tex]
After this step-by-step factorization, we conclude that the factored form is:
[tex]\[ \boxed{(x^3 - 10)\left(x^6 + 10x^3 + 100\right)} \][/tex]
Among the provided options, this matches with option B:
[tex]\[ B. (x^3 - 10)(x^6 + 10x^3 + 100) \][/tex]
So the correct answer is B.
1. Recognize that \( 1000 = 10^3 \):
[tex]\[ x^9 - 1000 = x^9 - 10^3 \][/tex]
2. See if it fits a known algebraic pattern: Here, we are dealing with the difference of cubes, since \( 10^3 \) is the cube of 10. The difference of cubes formula is:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
3. Rewrite the expression using the difference of cubes formula with \( a = x^3 \) and \( b = 10 \):
[tex]\[ x^9 - 10^3 = (x^3)^3 - 10^3 = (x^3 - 10)\left((x^3)^2 + (x^3)(10) + 10^2\right) \][/tex]
4. Simplify the quadratic term in the factorization:
[tex]\[ (x^3 - 10)\left(x^6 + 10x^3 + 100\right) \][/tex]
After this step-by-step factorization, we conclude that the factored form is:
[tex]\[ \boxed{(x^3 - 10)\left(x^6 + 10x^3 + 100\right)} \][/tex]
Among the provided options, this matches with option B:
[tex]\[ B. (x^3 - 10)(x^6 + 10x^3 + 100) \][/tex]
So the correct answer is B.
