Full solution

Q. Algebra Sp

2

2024

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Nork: Unit

4

Exam Review

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Divide using synthetic division.

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\begin{array}{l} \frac{x^{5}+11 x^{3}-12}{x-1} \\ \frac{x^{5}+11 x^{3}-12}{x-1}=\square+\frac{\square}{x-1} \end{array}

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(Simplify your answers. Do not factor.)

Set up synthetic division: Set up synthetic division with $1$ as the zero of $x - 1$ and the coefficients of the polynomial, including zeros for any missing terms.
Begin synthetic division: The coefficients are $1$ for $x^5$ , $0$ for $x^4$ , $11$ for $x^3$ , $0$ for $x^2$ , $0$ for $x$ , and $x^5$ $0$ for the constant term.
Multiply and add coefficients: Begin synthetic division: Bring down the leading coefficient $1$ .
Repeat multiplication and addition: Multiply the leading coefficient by the zero ( $1$ ) and write the result under the next coefficient ( $0$ ).
Complete synthetic division: Add the numbers in the second column to get the new second coefficient $0 + 1 = 1$ .
Read off quotient and remainder: Repeat the multiplication and addition process for the rest of the coefficients.
Write final answer: The synthetic division should look like this: $\newline$ $1 | 1\ 0\ 11\ 0\ 0\ -12$ $\newline$ $\phantom{1 |}\ \phantom{1}\ 1\ 1\ 12\ 12\ 12$ $\newline$ $\underline{\phantom{1 | 1 0 11 0 0 -12}}$ $\newline$ $\phantom{1 |}\ 1\ 1\ 12\ 12\ 12\ 0$
Write final answer: The synthetic division should look like this: $\newline$ $1 | 1 0 11 0 0 -12$ $\newline$ $| 1 1 12 12 12$ $\newline$ $---------------------$ $\newline$ $1 1 12 12 12 0$ Read off the result from the bottom row: The quotient is $x^4 + x^3 + 12x^2 + 12x + 12$ , and the remainder is $0$ .
Write final answer: The synthetic division should look like this: $\newline$ $1 | 1 0 11 0 0 -12$ $\newline$ $| 1 1 12 12 12$ $\newline$ $---------------------$ $\newline$ $1 1 12 12 12 0$ Read off the result from the bottom row: The quotient is $x^4 + x^3 + 12x^2 + 12x + 12$ , and the remainder is $0$ .Write the final answer in the form of a polynomial plus the remainder over the divisor.