Transcript Polynomial Operations PowerPoint

POLYNOMIAL OPERATIONS
INTEGRATED MATHEMATICS
TYPES OF POLYNOMIALS
Name
Monomial
Binomial
Trinomial
Polynomial
# Terms
Example
LIKE TERMS
• Same Variable
• Same Exponent
Example:
2
2𝑥
and
2
3𝑥
DEGREE: LARGEST EXPONENT ON POLYNOMIAL
• 5𝑥
3
5
2
3
2
• 3𝑥 + 4𝑥
• 5𝑥 + 2𝑥 + 4
DESCENDING ORDER: HIGHEST DEGREE FIRST
2
• 3𝑥
+
5
4𝑥
− 7𝑥
4
• 5𝑦 − 9 − 2𝑦 − 6𝑦
3
DESCENDING ORDER:
4 2
5 2
3 2
• 3𝑥 𝑦 + 2𝑥 𝑦 − 8𝑥 𝑦
3
• 3𝑥 𝑦
−
3
8𝑥𝑦
+
4
4
5𝑥 𝑦
TRY ON YOUR OWN
1.
𝟑
𝟐𝒙
− 𝟓𝒙 +
𝟒
𝟓
𝟕𝒙
2. 𝟒𝒙 𝒚 − 𝒙 + 𝟕𝒙
𝟑
𝟔
𝟒
3. 𝟐𝒙 − 𝟓𝒙𝒚 + 𝟕𝒙
𝟔
4. 4𝒂 + 𝟗𝒙𝒚𝒛 + 𝟕𝒙
𝟕
𝟓
EXAMPLE 1
2
2
3𝑥 + 2𝑥 − 2 + (−2𝑥 + 5𝑥 + 5)
EXAMPLE 2
31𝑚4 + 𝑚2 + 2𝑚 − 1 + (−7𝑚4 + 5𝑚2
− 2𝑚 + 2)
EXAMPLE 3
2
2
4𝑎 𝑏 − 5𝑎 + 2 + (−2𝑎 𝑏 − 2𝑎 − 4)
EXAMPLE 4
3
3 2
3
3 2
3𝑛 − 3𝑚 𝑛 − 5𝑛 − 3 + (5𝑛 + 2𝑚 𝑛 − 3𝑚 − 2𝑛 − 2)
EXAMPLE 5
3
2
4
2
−2𝑚 − 5𝑚 − 2𝑚 − 4 + (𝑚 − 6𝑚 + 7𝑚 − 10)
EXAMPLE 6
4
3
−2𝑥 𝑦
− 5𝑥𝑦 + 2 +
4
3
(𝑥 𝑦
+
2
𝑥
+ 2𝑥𝑦 + 5)
TRY ON YOUR OWN
2
2
1.
2𝑥 + 7𝑥 + 3 + (−5𝑥 − 3𝑥 − 6)
2.
𝟒
𝟒𝒙
3.
𝟑
𝟔𝒙
4.
𝟐
𝟑𝒙 𝒛
−𝒙+𝟕 +
−
𝟒
𝟑𝒙𝒚
−
𝟒
(𝟗𝒙
+𝟕 +
𝟒
𝟓𝒚 𝒙
+ 𝟑𝒙 − 𝟕)
𝟑
(𝟐𝒙
+ 𝟑𝒙 +
−
𝟒
𝟓𝒙𝒚
𝟐
−𝟐𝒙 𝒛 −
+
𝟕
𝟕𝒙 )
𝟒
𝒚 𝒙
− 𝟐𝒙
Subtract
EXAMPLE 7
3
2
4
3
2
𝑎 − 2𝑎 + 4 − (𝑎 − 4𝑎 − 3𝑎 )
EXAMPLE 8
4𝑥 3 𝑦 + 2𝑥 2 𝑦 2 − 3𝑥𝑦 + 6 − (𝑥 3 𝑦 − 2𝑥 2 𝑦 2 − 2𝑥𝑦 − 3)
EXAMPLE 9
3 2
3 2
2 2
3𝑥 𝑦 − 4𝑥𝑦 + 1 − (−4𝑥 𝑦 − 3𝑥 𝑦 + 3𝑥𝑦 − 5)
EXAMPLE 10
2
2
5𝑝 − 3𝑝 + 6 − (9𝑝 − 5𝑝 − 3)
EXAMPLE 11
3
2
3
2
4𝑥 + 2𝑥 − 2𝑥 − 3 − (2𝑥 − 3𝑥 + 2)
TRY ON YOUR OWN
2
2
1.
2𝑥 + 7𝑥 + 3 − (−5𝑥 − 3𝑥 − 6)
2.
𝟒
𝟒𝒙
3.
𝟑
𝟔𝒙
4.
𝟐
𝟑𝒙 𝒛
−𝒙+𝟕 −
−
𝟒
𝟑𝒙𝒚
−
𝟒
(𝟗𝒙
+𝟕 −
𝟒
𝟓𝒚 𝒙
+ 𝟑𝒙 − 𝟕)
𝟑
(𝟐𝒙
+ 𝟑𝒙 −
−
𝟒
𝟓𝒙𝒚
𝟐
−𝟐𝒙 𝒛 −
+
𝟕
𝟕𝒙 )
𝟒
𝒚 𝒙
− 𝟐𝒙
MULTIPLYING POLYNOMIALS
•
SIMPLIFY USING THE DISTRIBUTIVE PROPERTY
Example 1
2x ( 5x + 3 )
•
SIMPLIFY USING THE DISTRIBUTIVE PROPERTY
Example 2
𝟐
−𝟓𝒙 (𝟑𝒙 + 𝟓)
•
SIMPLIFY USING THE DISTRIBUTIVE PROPERTY
Example 3
𝟐
𝟑
𝟑𝒂 (−𝟓𝒂 + 𝟐𝒂 − 𝟕)
•
SIMPLIFY USING THE DISTRIBUTIVE PROPERTY
Example 4
𝟒
𝟑 𝟐
8p(𝟑𝒒 − 𝟐𝒒 𝒑 + 𝟐𝒑)
TRY ON YOUR OWN
1. 𝟑 𝒙 − 𝟕
𝟐
𝟐
2. 𝟐𝒙 (𝟓𝒙 − 𝟕𝒙 + 𝟏)
𝟐
3. −𝟖𝒚(−𝟐𝒚 −𝟕)
MULTIPLYING BINOMIALS
HOW WOULD YOU USE THE DISTRIBUTIVE
PROPERTY TO SIMPLIFY THIS?
( x + 1) ( x + 5 )
binomial
binomial
HOW WOULD YOU USE THE DISTRIBUTIVE
PROPERTY TO SIMPLIFY THIS?
Example 5
( x + 1) ( x + 5 )
Example 6
( 2n + 3) ( n - 6 )
Example 7
(𝒄 − 𝟓)(𝟑𝒄 + 𝟏)
Example 8
(𝒙 − 𝟏)(𝟑𝒙 − 𝟏)
TRY ON YOUR OWN
1. (𝟐𝒙 − 𝟓)(𝒙 + 𝟕)
5.
(𝒙 + 𝟐)(𝟒𝒙 + 𝟏)
2. (𝒙 + 𝟒)(𝟑𝒙 − 𝟏)
6. (𝒙 + 𝟓)(𝟐𝒙 − 𝟔)
3. (𝒃 − 𝟏)(𝒃 − 𝟗)
7. (𝟐𝒂 + 𝟑)(𝟑𝒂 + 𝟏)
4. 𝟔𝒙 + 𝟑 (𝟐𝒙 + 𝟕)
8. (𝟓𝒙 − 𝟓)(𝟖𝒙 + 𝟗)
SPECIAL CASES
Example 9
(𝒚 + 𝟓)(𝒚 − 𝟓)
SPECIAL CASES
Example 10
𝟐
𝟐
(𝟐𝒙 − 𝟏)(𝟑𝒙 + 𝟏)
SPECIAL CASES
Example 11
(𝟒𝒙 + 𝟕)(𝟒𝒙 − 𝟕)
SPECIAL CASES
Example 12
(𝒚 − 𝟓)
𝟐
SPECIAL CASES
Example 13
(𝟐𝒙 + 𝟑)
𝟐
SPECIAL CASES
Example 14
𝟐
(𝟐𝒗 − 𝟖)
𝟐
TRY ON YOUR OWN
1. (𝟐𝒙 − 𝟓)
2. (𝒙 − 𝟓)
𝟐
𝟐
3. (−𝟐𝒙 + 𝟕)
4. (𝟑𝒙 − 𝟏)
𝟐
𝟐
BINOMIALS & TRINOMIALS
Example 1
𝟐
(𝒙 + 𝟏)(𝒙 + 𝒙 + 𝟏)
BINOMIALS & TRINOMIALS
Example 2
𝟐
𝟐
(𝒙 + 𝟒)(𝒙 + 𝟐𝒙 − 𝟑)
BINOMIALS & TRINOMIALS
Example 3
𝟐
(𝟑𝒙 + 𝟐)(𝟑𝒙 + 𝟐𝒙 − 𝟏)
BINOMIALS & TRINOMIALS
Example 4
𝟐
(𝟑𝒄 − 𝟒)(𝟐𝒄 − 𝒄 + 𝟑)
TRY ON YOUR OWN
𝟐
1. (𝟐𝒕 − 𝟖)(𝟑𝒕 − 𝒕 + 𝟒)
2. (𝟓𝒗 +
3.
𝟐
(𝟒𝒅
𝟐
𝟐)(𝟑𝒗
+
+ 𝟐𝒗 − 𝟖)
𝟐
𝟑)(𝟐𝒅
𝟒
+ 𝒅 + 𝟓)
4. (𝟑𝒃 − 𝟕)(𝟓𝒃 − 𝟓𝒃 − 𝟗)