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SLIDE SHOW INSTRUCTIONS This presentation is completely under your control. This lesson will show only one step at a time, to see the next step you must press a key. (Actual names written on a key are in green ) TO STOP THE SLIDE SHOW : press ‘escape’ ( Esc , top left of keyboard) - PowerPoint PPT Presentation
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SLIDE SHOW INSTRUCTIONS
This presentation is completely under your control.This presentation is completely under your control.
This lesson will show only one step at a time,This lesson will show only one step at a time, to see the next step you must press a key. to see the next step you must press a key.
(Actual names written on a key are in (Actual names written on a key are in greengreen))
•TO STOP THE SLIDE SHOW: press ‘escape’ (Esc, top left of keyboard)
•TO MOVE FORWARD: press the “spacebar” or Enter(PageDn, , , also work)
•TO MOVE BACKWARD: press the key(PageUp, or also work)
Polynomial Addition:
Like Terms
To add polynomials, we must CombineCombine Like TermsLike Terms
Say we want to add these two polynomials:x2 - 3x + 4 and 5x2 - 2x - 2
x2 + 5x2
6x6x22
Like Terms have exactly the same variables with exactly the same powers.
Combine Like Terms by adding or subtracting the coefficients, but keep the variables (and powers) the same.
(Use the sign rules of integers to determine whether to add or subtract)
These terms both havex2, so they are like terms
To add polynomials, we must CombineCombine Like TermsLike Terms
Now add the next set of like terms:x2 - 3x + 4 and 5x2 - 2x - 2
-3x - 2x
6x6x22
Like Terms have exactly the same variables with exactly the same powers.
Combine Like Terms by adding or subtracting the coefficients, but keep the variables (and powers) the same.
(Use the sign rules of integers to determine whether to add or subtract)
- 5x- 5x
These terms each have an x, so they are like terms
To add polynomials, we must CombineCombine Like TermsLike Terms
Now add the last set of like terms:x2 - 3x + 4 and 5x2 - 2x - 2
+4 - 2
6x6x22
Like Terms have exactly the same variables with exactly the same powers.
Answer: 6x2 - 5x + 2
- 5x- 5x
These terms are constants(numbers with no variables),
so they are like terms
+ 2+ 2
Horizontal Method
One method of adding polynomials is called the
Horizontal MethodHorizontal Method
Add 2x2 - x - 7 and -x2 + 3x - 4, use the Horizontal Method:
Write the second polynomial in
parentheses with a plus sign between
them
2x2 - x - 7 + ( -x2 + 3x - 4)+ 1
Distribute the +1
2x2 - x - 7 - x2 +3x - 4
2x2 - x2
xx22
Combine Like Terms by adding or subtracting the coefficients, but keep the variables (and powers) the same.
(Use the sign rules of integers to determine whether to add or subtract)
These terms both havex2, so they are like terms
2x2 - x - 7 - x2 + 3x - 4
Horizontal Method: Horizontal Method: Polynomial Addition
Now add the first set of like terms:
Horizontal Method: Horizontal Method: Polynomial Addition
Now add the next set of like terms:2x2 - x - 7 - x2 + 3x - 4
xx22
-x + 3xThese terms each have an x, so they are like terms
+ 2x+ 2x
-7 - 4
xx22
Answer: x2 + 2x - 11
+ 2x+ 2x
These terms are constants(numbers with no variables), so they are like terms
- 11- 11
Horizontal Method: Horizontal Method: Polynomial Addition
Now add the next set of like terms:2x2 - x - 7 - x2 + 3x - 4
Practice Problems: (Hit enter to see the answers)Add using the Horizontal Method
1) -6x2 + 2x + 1 and 3x2 - x + 2 5) 5x + 2x - 3 and 4x + 2
2) 5xy + 4x and -3xy - 12x 6) -3y2 + 2y and y2 + y - 1
3) 4ab + 2a2b and 3ab 7) 2xy - 5x and - 3xy + 6x - 7
4) 3x2y +4x3y and - x3y + 2x2y 8) -17x + 6 and 3x - 6
Answers:1) -3x2 + x + 32) 2xy - 8x3) 2a2b + 7ab4) 3x3y + 5x2y5) 11x - 16) -2y2 + 3y - 17) -xy + x - 78) -14x
Vertical Method
Add 4x2 + 3x - 6 and 2x2 - 5x + 4, use the Vertical Method:
4x2
+ 2x2
Write the two polynomials so
that the like terms are stacked on
top of each otherThese terms both have
x2, so they are like terms
Vertical Method: Vertical Method: Polynomial Addition
4x2
+ 2x2
Write the two polynomials so that the like terms are stacked on top of each otherThese terms both have an
x, so they are like terms
+ 3x
- 5x
Add 4x2 + 3x - 6 and 2x2 - 5x + 4, use the Vertical Method:
Vertical Method: Vertical Method: Polynomial Addition
4x2
+ 2x2
Write the two polynomials so
that the like terms are stacked on
top of each otherThese terms are constants, so they are like terms
+ 3x
- 5x - 6
+ 4
Add 4x2 + 3x - 6 and 2x2 - 5x + 4, use the Vertical Method:
Vertical Method: Vertical Method: Polynomial Addition
Now draw a line under the whole thing and add the
coefficients.
4x2
+ 2x2
+ 3x
- 5x - 6
+ 4
6x2 - 2x - 2ANSWER = ANSWER =
Add 4x2 + 3x - 6 and 2x2 - 5x + 4, use the Vertical Method:
Vertical Method: Vertical Method: Polynomial Addition
Add x2 + 2 and 6x2 - 5x - 3, use the Vertical Method:
Write the two polynomials so
that the like terms are stacked on
top of each other
These terms both havex2, so they are like terms
x2
+ 6x2
Vertical Method: Vertical Method: Polynomial Addition
Add x2 + 2 and 6x2 - 5x - 3, use the Vertical Method:
x2
+ 6x2
+ 0x - 5x
(no x term?)
Solution: Solution: Write in a zero where there
are missing terms. (Or you can leave a blank spot)
A problem that comes up when using the Vertical
Method is that sometimes there are terms missing.
Vertical Method: Vertical Method: Polynomial Addition
Add x2 + 2 and 6x2 - 5x - 3, use the Vertical Method:
Write the two polynomials so
that the like terms are stacked on
top of each other
These terms are constants, so they are like terms
x2
+ 6x2
+ 2
- 3 + 0x - 5x
Vertical Method: Vertical Method: Polynomial Addition
Now draw a line under the whole thing and add the
coefficients.
7x2 - 5x - 1ANSWER = ANSWER =
- 5x
x2
+ 6x2
+ 0x + 2
- 3
Add x2 + 2 and 6x2 - 5x - 3, use the Vertical Method:
Vertical Method: Vertical Method: Polynomial Addition
Suggestions for other situations:Suggestions for other situations:SituationSituation SolutionSolution
1. A term has no coefficient showing Write a “1” in front of it
Example: x2 + 3x + 1 1x2 + 3x + 1
2. There are more than two like terms Stack (or group) all like terms together Ex: 2x + 6x - 3 and 4x + 5 (2x + 6x + 4x) + (-3 + 5)
3. There are many missing terms Write in zeros for each of themEx: 5x3 - 2x and 4x4 + 3x2 + x - 6 0x4 + 5x3 + 0x2 - 2x + 0
4x4 + 0x3 + 3x2 + 1x - 64x4 + 5x3 + 3x2 + 1x - 6
4. Subtraction problem Distribute the (-1) before working the problem.
x2 + 3x + 1 - (2x2 + 6x - 2) x2 + 3x + 1 - 2x2 - 6x + 2
Practice Problems: (Hit enter to see the answers)Add using the Vertical Method
1)-6x2 + 2x + 1 and 3x2 - x + 2 5) 5x + 2x - 3 and 4x + 2
2) 5y + 4x and -3y - 12x 6) -3y2 + 2y and y2 + y - 1
3) 4ab + 2a2 and 3ab 7) 2xy - 5x and - 3xy + 6x - 7
4) 3x2y +4xy and - xy + 2x2y 8) -17x2 + 6 and 3x - 6
Answers:1) -3x2 + x + 32) 2y - 8x3) 7ab + 2a2
4) 5x2y+ 3xy 5) 11x - 16) -2y2 + 3y - 17) -xy + x - 78) -17x2 + 3x
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