High School Math Solutions – Polynomials Calculator, Dividing Polynomials

In the last post, we talked about how to multiply polynomials. In this post, we will talk about to divide polynomials.  When we divide polynomials, we can write these problems in the form of a fraction. This allows us to reduce the fraction and get our answer. Although this method doesn’t work every time we divide polynomials, when it does, it is a quick, simple method.

Once written in the form of a fraction, there are two ways to reduce the fraction. One method is to split the fraction up, so there is one term per fraction, having the denominator be the same. The other method is to factor the numerator and denominator and cancel out terms that are the same in the numerator and denominator.

Let’s see some examples to better understand how to solve these problems.

We will use the splitting method first for our first example.

First example (click here):

(32x^4-56x^2)\div 8x

1. Rewrite the problem in the form of a fraction

\frac{32x^4-56x^2}{8x}

2. Split the fraction

\frac{32x^4}{8x}-\frac{56x^2}{8x}

3. Reduce the fractions

\frac{8\cdot 4x^4}{8x}-\frac{8\cdot7x^2}{8x}
4x^3-7x

Next example:
We will use our same problem above, but use factoring to solve the problem.

(32x^4-56x^2)\div 8x

1. Rewrite the problem in the form of a fraction

\frac{32x^4-56x^2}{8x}

2. Factor the numerator and denominator

\frac{8x^2(4x^2-7)}{8x}

3. Cancel out common factors

4. Simplify

x(4x^2-7)
4x^3-7x

Last example (click here):

(2x^3+11x^2+5x)\div(2x^2+x)

1. Rewrite the problem in the form of a fraction

\frac{2x^3+11x^2+5x}{2x^2+x}

2. Factor the numerator and denominator

\frac{(2x+1)(x+5)x}{(2x+1)x}

3. Cancel out common factors

4. Simplify

x+5

As you can see, factoring is a big part of dividing polynomials. Before attempting these problems, make sure you’ve mastered factoring. Dividing polynomials by these two methods is pretty simple. However, these methods may not work for certain division problems. Next blog post, we will talk about another method to use, when these methods don’t work. For more practice and help on this topic, checkout Symbolab’s Practice.

Until next time,

Leah

Middle School Math Solutions – Polynomials Calculator, Factoring Quadratics

Just like numbers have factors (2×3=6), expressions have factors ((x+2)(x+3)=x^2+5x+6). Factoring is the process of finding factors of a number or expression, where we find what multiplies together to make the expression or number.  In today’s blog post, we will talk about how to factor simple expressions and quadratics.

Factoring simple expressions – 
Given a simple expression, ax+b, pull out the greatest common factor from the expression. Pretty simple!
Here’s an example (click here):

Factor\:2x+6
2x+2\cdot3
2(x+3)

Factoring quadratics – 
Quadratics have the form: (ax)^2+bx+c, where a, b, and c are numbers.
Here are the steps for factoring quadratics:

1. Find u and v such that u∙v=a∙c and u+v=b
   This means u and v are factors of a∙c that when added together equal b
2. Rewrite the expression as (ax^2+ux)+(vx+c)
3. Factor out what you can from each parentheses
4. Factor out a common term
5. Check by multiplying the factors together (FOIL)

Here’s an example, where 1 is the leading coefficient (click here):

Factor\:x^2-5x+6

1. Find u and v such that u∙v=a∙c and u+v=b

a∙c=1∙6=6

6 can be written as the product of 1 and 6, -1 and -6, 3 and 2, or of -3 and -2. We need to pick the factors of 6 that equal -5. 

-3+(-2)=-5
-3∙-2=6
u=-3,\:v=-2

2. Rewrite the expression as (ax^2+ux)+(vx+c)

(x^2+(-3x))+(-2x+6)
(x^2-3x)+(-2x+6)

3. Factor out what you can from each parentheses

(x^2-3x)+(-2x+6)
x(x-3)+2(-x+3)
x(x-3)-2(x-3)

We factored out a -1 from (-x+3) on the last step because we want the expressions inside the parentheses to be the same for step 4.

4. Factor out a common term

x(x-3)-2(x-3)
(x-3)(x-2)

5 .Check by multiplying the factors together

(x-3)(x-2)=x^2-2x-3x+6=x^2-5x+6

Here’s an example when 1 isn’t the leading coefficient (click here):

Factor\:2x^2+x-6

1. Find u and v such that u∙v=a∙c and u+v=b

a∙c=2∙-6=-12

Factors of -12:    -12 and 1, 12 and -1, -6 and 2, 6 and -2, -4 and 3, 4 and -3
4-3=1
4∙-3=-12
u=-4,\:v=3

2. Rewrite the expression as (ax^2+ux)+(vx+c)

(2x^2-4x)+(3x-6)

3. Factor out what you can from each parentheses

(2x^2-4x)+(3x-6)
2x(x-2)+3(x-2)

4. Factor out a common term

(x-2)(2x+3)

5. Check by multiplying the factors together

(x-2)(2x+3)=2x^2+3x-4x-6=(2x)^2-x-6

The best advice for factoring quadratics is to practice factoring as many quadratics as you can. The more you practice factoring, the faster you will get. Soon, you’ll be able to skip the steps and factor it all in your head within seconds. For more help or practice on the topic, check out  Symbolab’s Practice.

Until next time,

Leah

Symbolab Study Groups…groups that work

stud•y group
noun
plural noun: study groups
a group of people who meet to study a particular subject and then report their findings or recommendations.

Study groups are great; more brainpower, boost motivation, support system.  But let’s be honest, it’s not always the most effective way to learn.  Sessions can turn into social events, schedule doesn’t work for everyone, we’re not always prepared for the sessions…

Symbolab Groups is a game changer.  There’s no better way to connect, share notes, and work through difficult problems together.  Symbolab Groups is a stress free study environment.  You get all the benefits of a study group minus the distractions.

Symbolab helps you stay focused.  No need to worry about missing out on group meetings, or not taking notes.  Stuck on a problem?  Just ask, your friends can help you instantly. Share problems, exercises and graphs, start a discussion, answer questions.

Symbolab Groups are there for you, always.

Cheers,
Michal