Up

Dividing

Honestly, this is definitely a tough subject. It can be hard to understand, so we will start slow and work up to it.

Dividing a polynomial by a monomial (one term) is a good place to start because it’s not that bad. Let’s say we have:

\(\Large \frac{{ - 12{x^3} + 6{x^2} + 9x}}{{3x}}\)

To be able to divide in this example, we need to divide each piece by 3x and reduce.

\(\Large \frac{{ - 12{x^3}}}{{3x}} + \frac{{6{x^2}}}{{3x}} + \frac{{9x}}{{3x}}\)

Answer: \( - 4{x^2} + 2x + 3\)

It becomes much tougher if we have to divide by a binomial (two terms). If this is the case, we have to do long division. Now, you’ll have to reach back in your memory to try to remember how to do long division. Let’s work though a basic example.

\(245 \div 7\)

Change the form to:

\(7\left){\vphantom{1{245}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{245}}}\)

Seven does not fit into 2, so we have to see how many times it will fit into 24 without going over. Seven times three works!

\(7\mathop{\left){\vphantom{1{245}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{245}}}}
\limits^{\displaystyle\,\,\, {3\_}}\)

\begin{matrix}
-21 &  \\
\hline
  & 35
  \end{matrix}
(bring down the 5)

Is this ringing a bell at all, yet? Let’s finish.

\(7\mathop{\left){\vphantom{1{245}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{245}}}}
\limits^{\displaystyle\,\,\, {35}}\)

\begin{matrix}
-21 &  \\
\hline
  & 35
  \end{matrix}

\begin{matrix}
-35 &  \\
\hline
  & 0
  \end{matrix}

Answer: 35

Let’s try to apply this to a polynomial.

\((2{x^3} - 8{x^2} + 25) \div (2x - 6)\)

Change to:

\(2x - 6\left){\vphantom{1{2{x^3} - 8{x^2} + 25}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{2{x^3} - 8{x^2} + 25}}}\)

You only have to worry about the 2x to start with. What can we multiply 2x by to match 2x3? The answer is x2.

\(2x - 6\mathop{\left){\vphantom{1{2{x^3} - 8{x^2} + 25}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{2{x^3} - 8{x^2} + 25}}}}
\limits^{\displaystyle\,\,\, {{x^2}\_\_\_\_\_\_\_\_}}\)

\begin{matrix}
  - {\text{2}}{x^{\text{3}}} - {\text{6}}{x^{\text{2}}} &\\
\hline
&  - {\text{ 2}}{x^{\text{2}}} + {\text{ 25}}
\end{matrix}

WORK:

x2(2x - 6) = 2x3 – 6x2

-8x2 – (-6x2) = -2x2

Now we want our 2x to match -2x2. We can multiply by –x.

\(2x - 6\mathop{\left){\vphantom{1{2{x^3} - 8{x^2} + 25}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{2{x^3} - 8{x^2} + 25}}}}
\limits^{\displaystyle \,\,\, {{x^2} - x\_\_\_\_\_}}\)

\begin{matrix}
  - {\text{2}}{x^{\text{3}}} - {\text{6}}{x^{\text{2}}} &\\
\hline
&  - {\text{ 2}}{x^{\text{2}}} + {\text{ 25}} \\
&  - { {\text{ }}\left( { - {\text{2}}{x^{\text{2}}}} \right){\text{ }} + {\text{ 6}}x} \\
\hline
& - {\text{6}}x + {\text{25}}
\end{matrix}

WORK:

-x(2x - 6)= -2x + 6x

6x and 25 cannot be combined since they are not “like terms,” so we leave it as 6x + 25. Almost done! To get 2x to match -6x we multiply by -3.

\(2x - 6\mathop{\left){\vphantom{1{2{x^3} - 8{x^2} + 25}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{2{x^3} - 8{x^2} + 25}}}}
\limits^{\displaystyle \,\,\, {{x^2} - x - 3\_\_}}\)

\begin{matrix}
  - {\text{2}}{x^{\text{3}}} - {\text{6}}{x^{\text{2}}} &\\
\hline
&  - {\text{ 2}}{x^{\text{2}}} + {\text{ 25}} \\
&  {- {\text{ }}\left( { - {\text{2}}{x^{\text{2}}}} \right){\text{ }} + {\text{ 6}}x} \\
\hline
& - {\text{6}}x + {\text{25}} \\
& -(- {\text{6}}x){\text{ }} + {\text{18}} \\
\hline
&{\text{7}}
\end{matrix}

WORK:

-3(2x – 6) = -6x

25 – (+18) = 7

There is no way to get our 2x to match 7 by multiplying so we leave the last piece of the answer in fraction form or \(\Large \frac{7}{{2x - 6}}\).

So our final answer is: \({x^2} - x - 3 + \Large \frac{7}{{2x - 6}}\).

Phew! That is a tough problem! Let’s try one more.

\((2{k^3} - 15{k^2} + 23k + 11) \div (2k - 5)\)

Change to:

\(2k - 5\left){\vphantom{1{2{k^3} - 15{k^2} + 23k + 11}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{2{k^3} - 15{k^2} + 23k + 11}}}\)

\(2k - 5\mathop{\left){\vphantom{1{2{k^3} - 15{k^2} + 23k + 11}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{2{k^3} - 15{k^2} + 23k + 11}}}}
\limits^{\displaystyle \,\,\, {{k^2}\_\_\_\_\_\_\_\_\_\_\_\_\_\_}}\)

\begin{matrix}
- {\text{ 2}}{k^{\text{3}}}-{\text{ 5}}{k^{\text{2}}} & \\
\hline
& - {\text{1}}0{k^{\text{2}}} + {\text{23}}k
\end{matrix}

\(2k - 5\mathop{\left){\vphantom{1{2{k^3} - 15{k^2} + 23k + 11}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{2{k^3} - 15{k^2} + 23k + 11}}}}
\limits^{\displaystyle \,\,\, {{k^2} - 5k\_\_\_\_\_\_\_\_\_}}\)

\begin{matrix}
- {\text{ 2}}{k^{\text{3}}}--{\text{ 5}}{k^{\text{2}}} & \\
\hline
& - {\text{1}}0{k^{\text{2}}} + {\text{23}}k \\
& - {\text{ }}\left( { - {\text{1}}0{k^2}} \right){\text{ }} + {\text{25}}k \\
\hline
& - {\text{2}}k + {\text{11}}
\end{matrix}

\(2k - 5\mathop{\left){\vphantom{1{2{k^3} - 15{k^2} + 23k + 11}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{2{k^3} - 15{k^2} + 23k + 11}}}}
\limits^{\displaystyle \,\,\, {{k^2} - 5k - 1\_\_\_\_\_\_\_}}\)

\begin{matrix}
- {\text{ 2}}{k^{\text{3}}}--{\text{ 5}}{k^{\text{2}}} & \\
\hline
& - {\text{1}}0{k^{\text{2}}} + {\text{23}}k \\
& - {\text{ }}\left( { - {\text{1}}0{k^2}} \right){\text{ }} + {\text{25}}k \\
\hline
& - {\text{2}}k + {\text{11}} \\
& - {\text{ }}\left( { - {\text{2}}k} \right) + {\text{ 5}} \\
\hline
& 6
\end{matrix}

Final answer: \({k^2} - 5k - 1 + \Large \frac{6}{{2k - 5}}\)

Below you can download some free math worksheets and practice.


Downloads:
1465 x

Divide.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Polynomials-Dividing-easy


Watch below how to solve this example:

 

Downloads:
1349 x

Divide.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Polynomials-Dividing-medium

Watch below how to solve this example:

 

Downloads:
1346 x

Divide.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Polynomials-Dividing-hard

Watch below how to solve this example:

 
 
 

Facebook PageGoogle PlusTwitterYouTube Channel

Algebra and Pre-Algebra

Beginning Algebra
Adding and subtracting integer numbers
Dividing integer numbers
Multiplying integer numbers
Sets of numbers
Order of operations
The Distributive Property
Verbal expressions
Beginning Trigonometry
Finding angles
Finding missing sides of triangles
Finding sine, cosine, tangent
Equations
Absolute value equations
Distance, rate, time word problems
Mixture word problems
Work word problems
One step equations
Multi step equations
Exponents
Graphing exponential functions
Operations and scientific notation
Properties of exponents
Writing scientific notation
Factoring
By grouping
Common factor only
Special cases
Linear Equations and Inequalities
Plotting points
Slope
Graphing absolute value equations
Percents
Percent of change
Markup, discount, and tax
Polynomials
Adding and subtracting
Dividing
Multiplying
Naming
Quadratic Functions
Completing the square by finding the constant
Graphing
Solving equations by completing the square
Solving equations by factoring
Solving equations by taking square roots
Solving equations with The Quadratic Formula
Understanding the discriminant
Inequalities
Absolute value inequalities
Graphing Single Variable Inequalities
Radical Expressions
Adding and subtracting
Dividing
Equations
Multiplying
Simplifying single radicals
The Distance Formula
The Midpoint Formula
Rational Expressions
Adding and subtracting
Equations
Multiplying and dividing
Simplifying and excluded values
Systems of Equations and Inequalities
Graphing systems of inequalities
Solving by elimination
Solving by graphing
Solving by substitution
Word problems