## Math 8 Chapter 1 Lesson 12: Sorted one-variable polynomial division

## 1. Theoretical Summary

**Eg: **Perform division:

\((2{x^5} + 3{x^3} + x):(2{x^2} + 1)\)

We do the following

- First we set the division:

\(\begin{array}{*{20}{c}}

{2{x^5} + 3{x^3} + x}\\

{\,\,\,}

\end{array}\left| {\begin{array}{*{20}{c}}

{2{x^2} + 1}\\

\hline

{\,\,\,}

\end{array}} \right.\)

- Then divide the highest degree term of the divisor by the highest degree term of the divisor:

\(2{x^5}:2{x^2} = {x^3}\)

- Multiply the quotient found for the divisor polynomial and then subtract the product of the divided polynomial to get the first remainder.

\(\begin{array}{*{20}{l}}

{2{x^5} + 3{x^3} + x}\\

{\underline {2{x^5} + {x^3}\,\,\,\,\,\,\,\,\,\,\,} }\\

{\,\,\,\,\,\,\,\,\,\,\,\,2{x^3} + x}\\

{}\\

{}

\end{array}\left| {\begin{array}{*{20}{c}}

{2{x^2} + 1}\\

\hline

{{x^3}}\\

{}\\

{}\\

{}

\end{array}} \right.\)

- Divide the term of the highest power of the first remainder by the term of the highest degree of the divisor, we get:

\(2{x^3}:2{x^2} = x\)

- Repeat the above steps, we get:

\(\begin{array}{*{20}{l}}

{2{x^5} + 3{x^3} + x}\\

{\underline {2{x^5} + {x^3}\,\,\,\,\,\,\,\,\,\,\,} }\\

{\,\,\,\,\,\,\,\,\,\,\,\,2{x^3} + x}\\

{\,\,\,\,\,\,\,\,\,\,\,\,\underline {2{x^3} + x} }\\

{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0 }

\end{array}\left| {\begin{array}{*{20}{c}}

{2{x^2} + 1}\\

\hline

{{x^3} + x}\\

{}\\

{}\\

{}

\end{array}} \right.\)

- Since the remainder is 0, the above division is divisible.

So the result of division \((2{x^5} + 3{x^3} + x):(2{x^2} + 1)\) is \({x^3} + x\)

**Note:**

- A division with a remainder of 0 is divisible.
- If the division has a non-zero remainder, we proceed in the same way until the highest power of the remainder is less than the highest power of the divisor polynomial.

## 2. Illustrated exercise

**Question 1: **Sort by the descending power of the variable and then do the division

\(\left( {x + 1 + 2{x^3} + {x^2}} \right):\left( {x – 1} \right)\)

**Solution guide**

– Sort by descending power of the variable, we get \(2{x^3} + {x^2} + x + 1\)

– Performing division we get

**Verse 2:** Perform the following division and determine the quotient and remainder:

\(\left( {2{x^3} – 3{x^2} + 6x – 4\,\,} \right):\,\,\left( {{x^2} – x + 3\ ,\,} \right)\,\)

**Solution guide**

So we find the quotient as \(2x-1\) and the remainder as \(-x-1\)

**Question 3: **Find the integer value of n so that A is divisible by B

\(A = 2{x^4} – {x^3} – {x^2} – x + n\,\,\,\,\,B = {x^2} + 1\)

**Solution guide**

Performing division we get

A is divisible by B \( \Leftrightarrow n – 3 = 0 \Leftrightarrow n = 3\)

So the value to look for is n = 3

### 3.1. Essay exercises

**Question 1: **Sort by the descending power of the variable and then do the division

\(\left( 2-7x+2{{x}^{3}}-{{x}^{2}} \right):\left( x-2 \right)\)

**Verse 2: **Perform the following division and determine the quotient and remainder

\(\left( 2{{x}^{3}}-5{{x}^{2}}+5x+1 \right):\left( {{x}^{2}}-3x+4 \right)\)

**Question 3: **Find the integer value of n so that A is divisible by B

\(A=2{{x}^{4}}+3{{x}^{3}}-12{{x}^{2}}-9x+n;\,\,B={{x) }^{2}}-3\)

### 3.2. Multiple choice exercises

**Question 1: **Sorting the following polynomial by the descending power of the variable gives which of the following results?

\({x^4} – 3 + 3{x^5} – 2{x^2} – {x^3}\)

A. \(3{x^5} + {x^4} – {x^3} – 2{x^2} – 3\)

B. \(3 – 2{x^2} – {x^3} + {x^4} + 3{x^5}\)

C. \({x^4} – 3 + 3{x^5} – 2{x^2} – {x^3}\)

D. \({x^4} – {x^3} + 3{x^5} – 2{x^2} – 3\)

**Verse 2: **The result of division \(\left( {{x^3} – {x^2} – 7x + 3} \right):\left( {x – 3} \right)\) is :

A. \({x^2} – 2x + 1\)

B. \({x^2} + 2x – 1\)

C. \({x^2} – x – 1\)

D. \({x^2} – x + 1\)

**Question 3: **For what value of n \(6{n^2} – n + 5\,\,\) is divisible by \(2n + 1\) with \(n \in Z\)

A. \(n \in \left\{ { – 1;3} \right\}\)

B. \(n \in \left\{ { – 1;0;3} \right\}\)

C. \(n \in \left\{ { – 4; – 3; – 1;0} \right\}\)

D. \(n \in \left\{ { – 4; – 1;0;3} \right\}\)

**Question 4: **Given \(A = {x^3} + {x^2} – 2x + 1\) and \(B = x – 1\) Find the remainder R and quotient Q in dividing A by B and then write A as A = BQ + R we get which of the following results?

A. \({x^3} + {x^2} – 2x + 1 = \left( {x – 1} \right)\left( {{x^2} + x} \right) + 3\)

B. \({x^3} + {x^2} – 2x + 1 = \left( {x – 1} \right)\left( {{x^2} + x} \right) + 1\)

C. \({x^3} + {x^2} – 2x + 1 = \left( {x – 1} \right)\left( {{x^2} + 2x} \right) + 3\)

D. \({x^3} + {x^2} – 2x + 1 = \left( {x – 1} \right)\left( {{x^2} + 2x} \right) + 1\)

**Question 5: **Let \(A = 8{x^2} – 26x + m\) and \(B = 2x – 3\) Find m so that A is divisible by B

A. m = 12

B. m = 14

C. m = 21

D.m = 41

## 4. Conclusion

Through this lesson, you should know the following:

- Know how to sort polynomials by ascending (descending) powers of variables.
- Perform polynomial division.
- Apply polynomial division to solve related problems.

.

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