Download 8-1 PPT

Monday: Announcements
• This is last week for 3rd grading period
• Today’s assignments are the last grades
• Unit 7 Reassessment sign-ups on the round table.
Deadline is Friday morning to reassess.
• Unit 8 is a 2 part unit (calculator and noncalculator)
• Afternoon tutorials this week start at 4:10pm
because of teacher duty
Add, Subtract,
Multiply, and Divide
Polynomials
Unit 8
Objectives
• I can add, subtract, and multiply polynomial
expressions.
• I can divide polynomial expressions using
Long Division
Polynomials
• A polynomial is a monomial or sum of
monomials.
• The monomials that make up the polynomial are
called terms of the polynomial.
• A trinomial has 3 unlike terms
• A binomial has 2 unlike terms
Standard Format
Exponents in descending ORDER
8 x  3x  3x  5 x  8 x  19
5
4
3
2
Standard Format
Exponents in descending ORDER
3x  2 x  3x  2 x  10 x  10
3
5
8
2
3x  2 x  3x  2 x  10 x  10
8
5
3
2
Addition and Subtraction
• Polynomials and monomials may only be
subtracted or added if they have like terms.
Addition and subtraction cannot be done on
unlike terms.
• To add or subtract, you just COMBINE
LIKE TERMS
Addition or Subtraction
•
•
•
•
Simplify the following:
(4x2 – 3x) – (x2 + 2x – 1)
4x2 – 3x – x2 - 2x + 1
3x2 – 5x + 1
Example #2
• Simplify: (3x + y) – (x + y) – (x + 3y)
• 3x + y – x - y – x - 3y
• x – 3y
Example 3
• Simplify (z2 – 6z – 10) + (2z2 + 4z – 11)
• z2 – 6z – 10 + 2z2 + 4z – 11
• 3z2 –2z - 21
EXAMPLE 1
Add polynomials vertically and horizontally
a. Add 2x3 – 5x2 + 3x – 9 and x3 + 6x2 + 11 in a vertical
format.
SOLUTION
a.
2x3 – 5x2 + 3x – 9
+ x3 + 6x2
+ 11
3x3 + x2 + 3x + 2
GUIDED PRACTICE
for Examples 1 and 2
Find the sum or difference.
1. (t2 – 6t + 2) + (5t2 – t – 8)
SOLUTION
t2 – 6t + 2
+ 5t2 – t – 8
6t2 – 7t – 6
Multiply with Distributive Prop
• Simplify: 3x(5x4 – x3 + 4x)
• 15x5 – 3x4 + 12x2
nd
2
Example:
• (x + 3)(x2 + 3x + 9)
• x3 + 3x2 + 9x + 3x2 + 9x + 27
• x3 + 6x2 + 18x + 27
Example 3
• Simplify: -3x2y(2x3y2 – 3x2y2 + 4x3y)
• -6x5y3 + 9x4y3 –12x5y2
Dividing Numbers
Quotient
When you divide a number
by another number and
there is no remainder:
Divisor
4
4 16
Then the divisor is a
factor!!
Dividend
Also the quotient becomes
another factor!!!
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Dividing Polynomials
Long division of polynomials is similar to long division of
whole numbers.
When you divide two polynomials you can check the answer
using the following:
dividend = (quotient • divisor) + remainder
The result is written in the form:
remainder
d iv id e n d  d iv is o r  quotient +
divisor
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Dividing Polynomials
Example: Divide x2 + 3x – 2 by x + 1 and check the answer.
2
x
1. x x 
x
x
2. x ( x  1 )  x 2  x
x + 2
2
x  1 x2  3x  2
x2 +
x
2x – 2
2x + 2
–4
3. ( x 2  3 x )  ( x 2  x )  2 x
4. x 2 x  2 x  2
x
5. 2 ( x  1 )  2 x  2
remainder
Answer: x + 2 +
6. ( 2 x  2 )  ( 2 x  2 )   4
–4
x 1
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Example: Divide 4x + 2x3 – 1 by 2x – 2 and check the answer.
x2 + x + 3
Write the terms of the dividend in
descending order.
2x  2 2x  0x  4x  1
3
2
2x3 – 2x2
Since there is no x2 term in the
dividend, add 0x2 as a placeholder.
2x2 + 4x
2x2 – 2x
6x – 1
6x – 6
5
Answer:
x2
+x+3
3
2
x
1.
2. x 2 ( 2 x  2 )  2 x 3  2 x 2
 x2
2x
2
2
x
3
3
2
2
3. 2 x  ( 2 x  2 x )  2 x
4.
x
2x
5. x ( 2 x  2 )  2 x 2  2 x
5
6. ( 2 x 2  4 x )  ( 2 x 2  2 x )  6 x
8. 3 ( 2 x  2 )  6 x  6
7. 6 x  3
2x
9. ( 6 x  1 )  ( 6 x  6 )  5  re m a in d e r
2x  2
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Example: Divide x2 – 5x + 6 by x – 2.
x – 3
x  2 x2  5x  6
x2 – 2x
– 3x + 6
– 3x + 6
0
Answer: x – 3 with no remainder.
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Homework
• WS 8-1