Divide the polynomial x^3−a^3 by x−a. Which polynomial identity does this establish?(1 point)
1. x^3-a^3=(x-a)(x^2+ax+a^2)
2. x^3-a^3=(x-a)(x^2-ax-a^2)
3. x^3-a^3=(x-a)(x^2+ax-a^2)
4. x^3-a^3=(x-a)(x^2-ax+a^2)
The correct polynomial identity established by dividing x^3−a^3 by x−a is:
1. x^3-a^3=(x-a)(x^2+ax+a^2)
To divide the polynomial x^3 - a^3 by x - a, we can use long division. Here's the step-by-step process:
Step 1: Write the polynomial division problem in long division format:
x^2 + ax + a^2
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x - a | x^3 + 0x^2 + 0x - a^3
Step 2: Divide the first term (x^3) by the divisor (x - a), giving x^2.
Step 3: Multiply the divisor (x - a) by the quotient from step 2 (x^2), giving x^3 - ax^2.
Step 4: Subtract the result from step 3 from the dividend (x^3 + 0x^2 + 0x - a^3):
(x^3 + 0x^2 + 0x - a^3) - (x^3 - ax^2)
= -ax^2 + 0x^2 + 0x - a^3
Step 5: Bring down the next term (-ax^2) and repeat the division process.
-ax^2 + ax + a^2
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x - a | -ax^2 + 0x^2 + 0x - a^3
Step 6: Divide the first term (-ax^2) by the divisor (x - a), giving -a.
Step 7: Multiply the divisor (x - a) by the quotient from step 6 (-a), giving -ax + a^2.
Step 8: Subtract the result from step 7 from the previous result:
(-ax^2 + 0x^2 + 0x - a^3) - (-ax + a^2)
= -ax^2 + ax - a^2 - a^3
Step 9: Bring down the next term (-ax) and repeat the division process.
-ax + a
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x - a | -ax^2 + ax - a^2 - a^3
Step 10: Divide the first term (-ax) by the divisor (x - a), giving -1.
Step 11: Multiply the divisor (x - a) by the quotient from step 10 (-1), giving -x + a.
Step 12: Subtract the result from step 11 from the previous result:
(-ax^2 + ax - a^2 - a^3) - (-x + a)
= -ax^2 + ax - a^2 + x - a^3 + a
Step 13: Simplify the terms:
-ax^2 + ax - a^2 + x - a^3 + a
= x - a^3 + (ax - ax^2) - (a^2 - a)
= x - a^3 + ax(1 - x) - a(1 - a)
= x - a^3 + ax(1 - x) - a(1 - a)
Therefore, the polynomial identity established by dividing x^3 - a^3 by x - a is:
x^3 - a^3 = (x - a)(x^2 + ax + a^2)
So, the correct answer is option 1: x^3 - a^3 = (x - a)(x^2 + ax + a^2).