Transcript Polynomial Addition

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Polynomial
Addition:
Like Terms
To add polynomials, we must Combine Like Terms
Say we want to add these two polynomials:
x2 - 3x + 4 and 5x2 - 2x - 2
x2 + 5x2
6x2
These terms both have
x2, so they are like terms
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)
To add polynomials, we must Combine Like Terms
Now add the next set of like terms:
x2 - 3x + 4 and 5x2 - 2x - 2
-3x - 2x
These terms each have an x,
so they are like terms
6x2 - 5x
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)
To add polynomials, we must Combine Like Terms
Now add the last set of like terms:
x2 - 3x + 4 and 5x2 - 2x - 2
Like Terms
These terms are constants
(numbers with no variables),
so they are like terms
+4 - 2
6x2 - 5x + 2
Answer: 6x2 - 5x + 2
have exactly the
same variables
with exactly the
same powers.
Horizontal
Method
One method of adding polynomials is called the
Horizontal Method
Add 2x2 - x - 7 and -x2 + 3x - 4, use the Horizontal Method:
Distribute
the +1
Write the second
polynomial in
2x2 - x - 7 ++1 ( -x2 + 3x - 4) parentheses with a
plus sign between
them
2x2 - x - 7 - x2 +3x - 4
Horizontal Method: Polynomial Addition
Now add the first set of like terms:
2x2 - x - 7 - x2 + 3x - 4
2x2 - x2
These terms both have
x2, so they are like terms
x2
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)
Horizontal Method: Polynomial Addition
Now add the next set of like terms:
2x2 - x - 7 - x2 + 3x - 4
-x + 3x
x2 + 2x
These terms each have an x,
so they are like terms
Horizontal Method: Polynomial Addition
Now add the next set of like terms:
2x2 - x - 7 - x2 + 3x - 4
These terms are constants
(numbers with no variables),
so they are like terms
-7 - 4
x2 + 2x - 11
Answer: x2 + 2x - 11
Practice Problems: (Hit enter to see the answers)
Add using the Horizontal Method
1) -6x2 + 2x + 1 and 3x2 - x + 2
5)
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 + 3
2) 2xy - 8x
3) 2a2b + 7ab
4) 3x3y + 5x2y
5) 11x - 1
6) -2y2 + 3y - 1
7) -xy + x - 7
8) -14x
5x + 2x - 3 and 4x + 2
Vertical
Method
Vertical Method: Polynomial Addition
Add 4x2 + 3x - 6 and 2x2 - 5x + 4, use the Vertical Method:
4x2
+ 2x2
These terms both have
x2, so they are like terms
Write the two
polynomials so
that the like terms
are stacked on
top of each other
Vertical Method: Polynomial Addition
Add 4x2 + 3x - 6 and 2x2 - 5x + 4, use the Vertical Method:
4x2 +
3x
+ 2x2 - 5x
These terms both have an
x, so they are like terms
Write the two
polynomials so
that the like terms
are stacked on
top of each other
Vertical Method: Polynomial Addition
Add 4x2 + 3x - 6 and 2x2 - 5x + 4, use the Vertical Method:
4x2 +
3x - 6
+ 2x2 - 5x + 4
These terms are constants,
so they are like terms
Write the two
polynomials so
that the like terms
are stacked on
top of each other
Vertical Method: Polynomial Addition
Add 4x2 + 3x - 6 and 2x2 - 5x + 4, use the Vertical Method:
4x2 +
3x - 6
+ 2x2 - 5x + 4
ANSWER =
6x2 - 2x - 2
Now draw a line
under the whole
thing and add the
coefficients.
Vertical Method: Polynomial Addition
Add x2 + 2 and 6x2 - 5x - 3, use the Vertical Method:
x2
+ 6x2
These terms both have
x2, so they are like terms
Write the two
polynomials so
that the like terms
are stacked on
top of each other
Vertical Method: Polynomial Addition
Add x2 + 2 and 6x2 - 5x - 3, use the Vertical Method:
x2 +(no0xx term?)
+ 6x2 - 5x
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: Polynomial Addition
Add x2 + 2 and 6x2 - 5x - 3, use the Vertical Method:
x2+ 0x + 2
+ 6x2 - 5x - 3
These terms are constants,
so they are like terms
Write the two
polynomials so
that the like terms
are stacked on
top of each other
Vertical Method: Polynomial Addition
Add x2 + 2 and 6x2 - 5x - 3, use the Vertical Method:
x2 +
0x + 2
+ 6x2 - 5x - 3
ANSWER =
7x2 - 5x - 1
Now draw a line
under the whole
thing and add the
coefficients.
Suggestions for other situations:
Situation
1. A term has no coefficient showing
Example: x2 + 3x + 1
2. There are more than two like terms
Ex: 2x + 6x - 3 and 4x + 5
3. There are many missing terms
Ex: 5x3 - 2x and 4x4 + 3x2 + x - 6
Solution
Write a “1” in front of it
1x2 + 3x + 1
Stack (or group) all like terms together
(2x + 6x + 4x) + (-3 + 5)
Write in zeros for each of them
0x4 + 5x3 + 0x2 - 2x + 0
4x4 + 0x3 + 3x2 + 1x - 6
4x4 + 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)
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 + 3
2) 2y - 8x
3) 7ab + 2a2
4) 5x2y+ 3xy
5) 11x - 1
6) -2y2 + 3y - 1
7) -xy + x - 7
8) -17x2 + 3x
5x + 2x - 3 and 4x + 2
End of Tutorial
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