Divide x⁴ + 2x³ + 2x² + 2x + 2 by (x² − 1), and write down the quotient and the remainder.
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Step-by-Step Solution
Step 1: Set up for polynomial long division
We need to perform polynomial long division. First, we write the dividend and the divisor in the standard long division format. Ensure all powers of x are present in the dividend, using zero coefficients if necessary.
Step 2: Perform the first step of division
Divide the leading term of the dividend (x4) by the leading term of the divisor (x2) to get x2. Multiply the divisor by x2 and subtract the result from the dividend. Bring down the remaining terms.
Step 3: Perform the second step of division
Now, divide the leading term of the new polynomial (2x3) by the leading term of the divisor (x2) to get 2x. Multiply the divisor by 2x and subtract the result. Bring down the next term.
Step 4: Perform the third step of division
Finally, divide the leading term of the current polynomial (3x2) by the leading term of the divisor (x2) to get 3. Multiply the divisor by 3 and subtract the result. The degree of the remaining polynomial (4x+5) is less than the degree of the divisor (x2−1), so this is our remainder.
Step 5: State the quotient and remainder
From the long division process, we have determined the quotient and the remainder.