Bot GPT 4-o mini To perform synthetic division of the polynomial \( x^3 - 4 \) by \( x + 2 \), we first note that synthetic division requires us to use the root of the divisor. The polynomial \( x + 2 \) has a root at \( x = -2 \).
Here are the steps to perform synthetic division:
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Identify the coefficients of the polynomial \( x^3 - 4 \). Since it's a cubic polynomial, we write it as \( 1x^3 + 0x^2 + 0x - 4 \). The coefficients are:
- 1 (for \( x^3 \))
- 0 (for \( x^2 \))
- 0 (for \( x^1 \))
- -4 (for the constant term)
So the coefficients are: \( [1, 0, 0, -4] \).
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Set up the synthetic division using the root \( -2 \):
-2 | 1 0 0 -4 | |________________ -
Bring down the leading coefficient:
-2 | 1 0 0 -4 | |________________ 1 -
Multiply \( -2 \) by the value just brought down (which is 1) and write the result under the next coefficient:
-2 | 1 0 0 -4 | -2 |________________ 1 -2 -
Now, add the values in the second column:
-2 | 1 0 0 -4 | -2 |________________ 1 -2 0 -
Repeat this process for the next column. Multiply \( -2 \) by \( -2 \) (which gives \( 4 \)) and write it under the next coefficient (which is 0):
-2 | 1 0 0 -4 | -2 4 |________________ 1 -2 4 -
Now add the values in the third column:
-2 | 1 0 0 -4 | -2 4 |________________ 1 -2 4 0 -
Finally, multiply \( -2 \) by \( 4 \) (which gives \( -8 \)) and add that to the last coefficient:
-2 | 1 0 0 -4 | -2 4 -8 |________________ 1 -2 4 -12
The result of the synthetic division is the polynomial \( x^2 - 2x + 4 \) plus a remainder of \( -12 \). Putting this all together, we can express the result of the division as:
\[ x^3 - 4 = (x + 2)(x^2 - 2x + 4) - 12 \]
So, in summary, the result of the synthetic division is:
\[ x^3 - 4 = (x + 2)(x^2 - 2x + 4) - 12 \]
Thus, \( (x^3 - 4) \div (x + 2) = x^2 - 2x + 4 \) with a remainder of \( -12 \).
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