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Math  /  Algebra

QuestionUse synthetic division to find the quotient. Specify the quotient and the remainder. If there is no remainder, enter a 0 in the appropriate answer box. (x4−2x2+4)÷(x+1)\left(x^{4}-2 x^{2}+4\right) \div(x+1) ++ □\square □\square x+1‾\overline{x+1}

Studdy Solution

STEP 1

1. We are dividing x4−2x2+4x^4 - 2x^2 + 4 by x+1x + 1.
2. We need to find the quotient and the remainder.
3. We will use synthetic division to solve this problem.

STEP 2

1. Set up the synthetic division.
2. Perform the synthetic division.
3. Interpret the results to find the quotient and remainder.

STEP 3

Set up the synthetic division: - Write -1 (the negative of the constant term in the divisor) at the top. - Write the coefficients of the dividend in descending order of degree, using 0 for missing terms.
−110−204\begin{array}{r} -1 \\ 1 \quad 0 \quad -2 \quad 0 \quad 4 \\ \hline \end{array}

STEP 4

Perform the synthetic division: - Bring down the first coefficient (1). - Multiply the result by -1 and write it below the next coefficient. - Add the numbers in this column. - Repeat the process for each column.
−110−204−111−11−1−113\begin{array}{r} -1 \\ 1 \quad 0 \quad -2 \quad 0 \quad 4 \\ \quad -1 \quad 1 \quad 1 \quad -1 \\ \hline 1 \quad -1 \quad -1 \quad 1 \quad 3 \\ \end{array}

STEP 5

Interpret the results: - The numbers in the bottom row represent the coefficients of the quotient, from highest degree to lowest, followed by the remainder. - The quotient is x3−x2−x+1x^3 - x^2 - x + 1. - The remainder is 3.
Therefore, the solution is:
Quotient: x3−x2−x+1x^3 - x^2 - x + 1 Remainder: 3
To fill in the blanks in the given format:
+3x3−x2−x+1+3 \quad x^3 - x^2 - x + 1 x+1‾\overline{x+1}

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