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Adding and Subtracting Polynomials

A step by step guide on how to add and subtract polynomials
by

Sophie L.

on 16 June 2015

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Transcript of Adding and Subtracting Polynomials

Obstacle 1
How to Subtract Polynomials
Word Problems
Conclusion
Start
Adding and Subtracting
Polynomials

By Sophie Liu
Algebra
6/15/15

How to Combine Like Terms?
How to Add Polynomials
Adding and Subtracting Polynomials
Adding and subtracting polynomials is a way of combining like terms (monomials or polynomials), with some order of operations thrown into it. There are two ways to add or subtract polynomials. You can use the horizontal method or use the vertical method.
Vocabulary
Monomial
- a number, a variable, or a product of a number and one or more variables with whole number exponents.

Polynomial
- a monomial or a sum of monomials. Each monomial is called a
term
of the polynomial.

Degree of a polynomial
- greatest degree of its terms. A polynomial is written in standard form when the exponents of the terms decrease from highest to lowest, left to right.

Binomial
- polynomial with two terms

Trinomial
- polynomial with three terms

T
erm
- the parts of an expression that are added together

Like terms
- terms that have identical, variable parts

How to Combine Like Terms
Before adding or subtracting polynomials, it's important to learn how to combine like terms. The key to combining like terms is to understand what like terms are and be able to identify which pair of terms is a pair of
like
terms.

For example, the terms 2x, 3x, -89x, and x are like terms because each term consists of a single variable, x, and a coefficient. If the variable wasn't x, then it wouldn't be like terms. The variable has to be x. Also, the exponent has to be the same integer.

Another example is 3y2, y2, -y2, 26y2. These monomials have the same vairable and the same exponent.

For comparison, below are a few examples of unlike terms.

The terms 56x and 7y are not like terms because the terms don't have the same variable.

The monomials 15y, 19y2, 31y5 are not alike because the exponents are not the same.

19x and 44xy are not the same because although both terms have a x variable, one term has a y variable thus these terms are not alike.
Adding Polynomials using the Vertical Method
When adding polynomials using the vertical method, you simply align the polynomials vertically and add the like terms.

An example is
(3a^2+8) + (5a-1)
.

1.) Align the polynomials vertically, making sure the like terms are aligned on top of one another.

2.) The problem should look like this:

3a^2 + 8
+

5a - 1
Leave a space for the missing term since 5a can't combine with anything else.
_______________

3.) Combine like terms, being careful of the negative sign. 3a^2 can't combine with anything else. The same goes for 5a. You can add 8 to (-1).

4.) The equation should look like this:
3a^2 +5a+7
.



equation in simplest form.
Remember to write the

Adding Polynomials Using the Horizontal Method
Adding polynomials using the horizontal method is simply combining like terms in the equation. Here is an example:

1.) Start to add like terms.

2.)


3.) The final equation is






(-x^2+5x+4)+(3x^2-8x+9).

-x^2 combines with 3x^2 to get 2x^2. 5x combines with -8x to get -3x. 4 combines with 9 to get 13.

(2x^2-3x+13).


If you are adding two
negatives, remember to
make the term a negative.

How to Subtract Polynomials Using the Vertical Method
Subtracting Polynomials using the vertical method is the same as adding polynomials using the vertical method. It's just important to remember the correct use of negatives. Align the polynomials vertically and subtract the like terms.

An example problem is
(y^2+4y+2) - (2y^2-5y-3)
.

1.) Align the polynomials vertically, making sure the like terms are on top of one another. Leave a space for the term that doesn't have another term that is alike with it.

2.) The equation should look like this:

y^2+4y+2
- 2y^2-5y-3
___________________

3.) Combine like terms, watching out for the negative signs. y^2 minus 2y^2 equals -y^2. 4y minus (-5y) equals 9y. 2 minus (-3) equals 5.

4.) The final equation should look like this:
-y^2+9y+5
.






Remember that two negatives equal a positive.
Watch out for

the negative sign.
How to Subtract Polynomials Using the Horizontal Method
Subtracting polynomials is combing like terms just like adding polynomials. Except of adding, you subtract the terms. Here is an example: .

1.) Start to subtract like terms.

2.)

3.) The final equation is
(5x^2+4x+1) - (2x^2-6)
Subtract 2x^2 from 5x^2. Subtract -6 from 1. Since no other term is a like term with 4x, you leave 4x by itself. 4x minus 0 is 4x, so you leave 4x as a positive.
(3x^2+4x+7).
Remember that 1-(-6)
comes out as positive 7.
When adding and subtracting polynomials, remember to include the negative/positive sign inside and outside of the parenthesis.
Word Problem: Adding Polynomials #1
Two airplanes depart from the same airport, traveling in opposite directions. After 2 hours, one airplane is x^2+ 2x+ 400 miles away
from the airport, and the other airplane is 3x^2+ 50x+ 100 miles away from the airport. What is the total distance between the two airplanes?

1.)To solve this problem, you can draw a picture to visualize the problem.

2.) Decide the operation needed to solve the problem. The operation is addition.

3.) By adding the two polynomials, you can find the total distance between the two
airplanes.

4.) The expression looks like this: (x^2+2x+400) + (3x^2+50x+100).

5.) You can add the polynomials using either the horizontal method or vertical
method.

6.) The final answer is (4x^2+52x+500) miles.
Airport- Starting distance: 0
Distance: x^2+2x+400
Distance: 3x^2+50x+100
Word Problem: Adding Polynomials #2
Billy Bob loves to draw pictures of rainbows and sunshine. He wants to know the perimeter of his painting (in inches) so he can find the perfect picture frame to put the painting in.

1.) Draw a diagram to help you determine the dimensions.

2.) The length is 14x^3+45x+78, and the width is 33x^4+77x+9.

3.) To find perimeter, you use the equation 2l+2w.

4.) The two operations used to solve this problem is multiplication and addition.

5.) The expression would look like this 2(14x^3+45x+78) + 2(33x^4+77x+9). You use the distributive property to simply the two trinomials. The simplified expressions are 28x^3+90x+156 and 66x^4+154x+18.

6.) You now add (28x^3+90x+156) and (66x^4+154x+18) using the horizontal method or vertical method, remembering to group only the like terms. The final answer is
(66x^4+28x^3+244x+174) inches.



14x^3+45x+78
33x^4+77x+9
Word Problem: Subtracting Polynomials #1
The area of a rectangle (shaded region) is 2a^2-4a+5 cm squared. The area of the square(non-shaded region) inside the rectangle is a^2-2a-6 cm squared. What is the area of the shaded region?

1.) Draw a diagram to visualize the problem.

2.) Since you are trying to find the shaded region only, you would use
subtraction as the operation needed to solve the problem.

3.) You would subtract the area of the square from the area of the
rectangle to take away the non-shaded region. This way, you will be left with the shaded region.

4.) The equation will look like this: (2a^2-4a+5)- (a^2-2a-6).

5.) Either using the vertical method or horizontal method, subtract the polynomials.

6.) The final answer is
(a^2-2a+11) cm. squared
.
2a^2-4a+5
a^2-2a-6
Word Problem: Subtracting Polynomials #2
Miranda is mass producing x amount of hats that is represented by the expression 5x+400,000. The revenue from the sales is represented by the equation is 0.00085x^2+20x. Find the polynomial expression for the profit of creating and selling hats. Evaluate the expression when x=200,000.

1.) Because revenue is the income generated from sales, you would subtract the cost of making the hats from the revenue to find the profit.

2.)
Revenue- cost= profit

3.) You write the equation like this
:
(0.00085x^2+20x)- (5x+400,000)
.

4.)Either using the vertical method or horizontal method, subtract like terms.

5.) The final answer is

0.00085x^2+15x-400,000
.

6.) The second part to the problem is finding the profit when Miranda is making 200,000 hats.

7.) Substitute x for 200,000.

8.) 0.00085(200,000)^2+15(200,000)-400,000

9.)The profit is $36,600,000.
Wrap-Up: Why Polynomials?
Adding and subtracting polynomials is just like adding and subtracting any normal equation, except you combine like terms.
The commutative property can be applied when adding polynomials.
When using a horizontal method to add/subtract like terms:
-Remove parentheses. Identify like terms. Group the like terms together. Add the like terms.
When using a vertical method to add/subtract like terms:
-Arrange the like terms so that they are lined up under one another in vertical columns, adding 0 place holders if necessary. Add the like terms in each column following the rules for adding/subtracting like numbers.
Polynomials are one of the most important and basic functions in mathematics.
Polynomials are used in quadratics and real life applications so it's important to understand what polynomials are.
For example, if an apple is thrown in the air, it makes a parabola which is a polynomial equation.
Scientists, mathematicians, engineers, or economists use polynomials to solve equations.
Polynomials appear in functions and physics problems.
Even though you may not see these types of problems everyday, that does not mean they do not have a use in your everyday life.
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