Fill in each blank with the appropriate response. x+3 x - 1 ) x ^ 2 + 2 x + 3

Name: Problem 4, Chapter 11: Algebra for College Students
Uploaded: 2021-05-07T18:29:49-08:00
Duration: 4 min 12 s
Channel: Ruby P
Description: Fill in each blank with the appropriate response. x+3 x - 1 ) x ^ 2 + 2 x + 3 x^2-x 3 x+3 3 x-3 6 can be written x^2+2 x+3= (x-1) () +

step 1 This question is from polynomials and rational functions. Here in this question, see, in algebra

Question

Fill in each blank with the appropriate response. $$ \begin{array}{c} \quad \quad\quad\quad\quad x+3 \\ x - 1 )\overline{ x ^ { 2 } + 2 x + 3 } \\ x^{2}-x \\\hline \quad\quad\quad\quad{3 x}+3 \\ \quad\quad\quad\quad 3 x-3\\ \hline \quad\quad\quad\quad\quad\quad{6} \end{array} $$ can be written $x^{2}+2 x+3=$ $(x-1)$ () + _

   Fill in each blank with the appropriate response.
$$
\begin{array}{c}
\quad \quad\quad\quad\quad x+3 \\
 x - 1 )\overline{ x ^ { 2 } + 2 x + 3 } \\
x^{2}-x \\\hline \quad\quad\quad\quad{3 x}+3 \\
\quad\quad\quad\quad 3 x-3\\ \hline \quad\quad\quad\quad\quad\quad{6}
\end{array}
$$
can be written $x^{2}+2 x+3=$ $(x-1)$ (____) + _____

Algebra for College Students

Margaret L. Lial,… 9th Edition

Chapter 11, Problem 4

Instant Answer

Solved by Expert Ruby P

05/07/2021

Step 1

First, we have the dividend $x^2 + 2x + 3$. Show more…

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Fill in each blank with the appropriate response. $$ \begin{array}{c} \quad \quad\quad\quad\quad x+3 \\ x - 1 )\overline{ x ^ { 2 } + 2 x + 3 } \\ x^{2}-x \\\hline \quad\quad\quad\quad{3 x}+3 \\ \quad\quad\quad\quad 3 x-3\\ \hline \quad\quad\quad\quad\quad\quad{6} \end{array} $$ can be written $x^{2}+2 x+3=$ $(x-1)$ (____) + _____

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Key Concepts

Remainder Theorem

The Remainder Theorem establishes that when a polynomial is divided by a linear divisor of the form (x - c), the remainder is equal to the value of the polynomial evaluated at c. This provides a quick way to find remainders and check divisibility.

Polynomial Division Algorithm

This concept states that for any dividend polynomial and any non-zero divisor polynomial, there exist unique polynomials called the quotient and the remainder such that the dividend equals the divisor multiplied by the quotient plus the remainder, where the degree of the remainder is less than the degree of the divisor.

Polynomial Long Division

This is a systematic method for dividing a polynomial by another polynomial of lower or equal degree. It resembles numerical long division and is used to determine the quotient and remainder, allowing for the rewriting of the original polynomial based on the division algorithm.

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