Answer :
To factor the expression \( x^9 - 1000 \), we can look for its factors by recognizing familiar patterns or applying algebraic identities.
Here's how we can approach it step-by-step:
1. Identify the expression: The expression to be factored is \( x^9 - 1000 \).
2. Pattern Recognition: We see that \(1000 = 10^3\), which suggests that \( x^9 - 1000 = x^9 - 10^3 \).
3. Factor using the difference of cubes:
Recall the algebraic identity for the difference of cubes:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
However, in our case, the exponent is 9, which implies we need to factor in steps.
4. Group terms:
Recognize that \( x^9 \) can be written as \( (x^3)^3 \). So we rewrite the expression as:
[tex]\[ x^9 - 1000 = (x^3)^3 - 10^3 \][/tex]
5. Apply the difference of cubes identity:
Using \( a = x^3 \) and \( b = 10 \),
[tex]\[ (x^3)^3 - 10^3 = (x^3 - 10)((x^3)^2 + x^3 \cdot 10 + 10^2) \][/tex]
6. Simplify inside the parentheses:
[tex]\[ = (x^3 - 10)(x^6 + 10x^3 + 100) \][/tex]
So, the factored form of \( x^9 - 1000 \) is:
[tex]\[ (x^3 - 10)(x^6 + 10x^3 + 100) \][/tex]
Now, let’s match this with the given multiple-choice options:
A. \(\left(x^3+10\right)\left(x^6-10 x^3+100\right)\)
B. \(\left(x^3-10\right)\left(x^6+10 x^3+100\right)\)
C. \((x-10)(x+10)\left(x^3-10 x^2+100\right)\)
D. \(\left(x^3-10\right)\left(x^3+10\right)\left(x^2-10 x+100\right)\)
The correct answer is:
[tex]\[ \boxed{\text{B. } \left(x^3-10\right)\left(x^6+10 x^3+100\right)} \][/tex]
Here's how we can approach it step-by-step:
1. Identify the expression: The expression to be factored is \( x^9 - 1000 \).
2. Pattern Recognition: We see that \(1000 = 10^3\), which suggests that \( x^9 - 1000 = x^9 - 10^3 \).
3. Factor using the difference of cubes:
Recall the algebraic identity for the difference of cubes:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
However, in our case, the exponent is 9, which implies we need to factor in steps.
4. Group terms:
Recognize that \( x^9 \) can be written as \( (x^3)^3 \). So we rewrite the expression as:
[tex]\[ x^9 - 1000 = (x^3)^3 - 10^3 \][/tex]
5. Apply the difference of cubes identity:
Using \( a = x^3 \) and \( b = 10 \),
[tex]\[ (x^3)^3 - 10^3 = (x^3 - 10)((x^3)^2 + x^3 \cdot 10 + 10^2) \][/tex]
6. Simplify inside the parentheses:
[tex]\[ = (x^3 - 10)(x^6 + 10x^3 + 100) \][/tex]
So, the factored form of \( x^9 - 1000 \) is:
[tex]\[ (x^3 - 10)(x^6 + 10x^3 + 100) \][/tex]
Now, let’s match this with the given multiple-choice options:
A. \(\left(x^3+10\right)\left(x^6-10 x^3+100\right)\)
B. \(\left(x^3-10\right)\left(x^6+10 x^3+100\right)\)
C. \((x-10)(x+10)\left(x^3-10 x^2+100\right)\)
D. \(\left(x^3-10\right)\left(x^3+10\right)\left(x^2-10 x+100\right)\)
The correct answer is:
[tex]\[ \boxed{\text{B. } \left(x^3-10\right)\left(x^6+10 x^3+100\right)} \][/tex]
