6.4 Multiplying/Dividing Polynomials 1/10/2014. How do you multiply 1256 by 13?
-
Upload
ashlyn-cobb -
Category
Documents
-
view
221 -
download
2
Transcript of 6.4 Multiplying/Dividing Polynomials 1/10/2014. How do you multiply 1256 by 13?
6.4 Multiplying/Dividing Polynomials
1/10/2014
How do you multiply 1256 by 13?
Example 1 Multiply Polynomials Vertically
Find the product .( )x 2 4x 7–+ ( )2x –
SOLUTION
Line up like terms vertically. Then multiply as shown below.
x 2 4x 7–+
2x –×
2x 2 8x + 14–– Multiply by 2.x 2 4x 7–+ –
x 3 7x+ –4x 2 Multiply by x.x 2 4x 7–+
x 3 15x+ –2x 2 + 14 Combine like terms.
Example 2 Multiply Polynomials Horizontally
Find the product.
( )4+3x ( )5x 2 x 6–+a.
4+3x( )5x 2 x 6–+ ( )5x 2 x 6–+= Use distributive property.
SOLUTION
( )4+3x ( )5x 2 x 6–+a.
+15x 3 18x–+ 20x 2 4x 24–+= 3x 2 Use distributive property.
15x 3 + 24–= 20x 23x 2 + 18x 4x– + Group like terms.
15x 3 14x–+= 23x 2 24– Combine like terms.
Example 2 Multiply Polynomials Horizontally
( )2x – ( )x 2 2x 3–+=
b. ( )2x –( )1x – ( )3x +
+x 3 3x–+ 2x 2 4x 6–= 2x 2 – Use distributive property.
x 3 + 6= 2x 22x 2 3x 4x–– – + Group like terms.
+x 3 7x– 6= Combine like terms.
2x –( )x 2 2x 3–+= ( )x 2 2x 3–+
𝑥2+2𝑥− 3 Multiply any 2 binomials
Multiply by the 3rd binomial
Checkpoint Multiply Polynomials
Find the product. Use either a horizontal or vertical format.
1. ( )1+x( )x 2 x 2+ +
2. ( )2x 2 x 4– + ( )3x –
ANSWER 2x 2 3x 2x 3 + + +
ANSWER 7x 2 7x 122x 3 +– –
Checkpoint Multiply Polynomials
Find the product. Use either a horizontal or vertical format.
3. ( )1+2x( )3x 2 x 1–+
ANSWER 5x 2 x 16x 3 + ––
ANSWER x 2 10x 8x 3 + – +
( )1x –4. ( )4x + ( )2x –
Checkpoint Use Special Product Patterns
Find the product.
( )7z –( )7z +5.
ANSWER z 2 49–
6. ( )223y +
12y9y 2 4+ +ANSWER
64x 3 +– 12x 48x 2 – 1ANSWER
7. –( )314x
Homework:
Worksheet 6.4
6.4 Multiplying/Dividing Polynomials….cont’d.
1/29/2014
Example 4 Use Long Division
Find the quotient 985 23.÷
Divide 98 by 23.98523-92 Subtract the product .4( )23 = 92
65 Bring down 5. Divide 65 by 23.
19 Remainder
ANSWER The result is written as .2319
42
-46 Subtract the product .2( )23 = 46
4 2
Example 5 Use Polynomial Long Division
Find the quotient .( )4x +x 3 + –6x 3x 2 – 4 ( ) ÷
Write division in the same format you use to divide whole numbers.
x 3 + 4x 2 Subtract the product .( )4x +x 2 = x 3 4x 2+
– 6x x 2– Bring down - 6x. Divide –x2 by x
– 4x x 2– Subtract the product . ( )4x +x = x 2 4x – – –
– 2x – 4 Bring down - 4. Divide -2x by x
4 Remainder
x 3 + – 6x 3x 2 – 4 x + 4 x 3 ÷ x = x 2
ANSWER
The result is written as .x 2 – –x 2x + 4
4+
– 2x – 8 Subtract the product ( )4x +2 = 2x 8– – – .
x2 -x -2
- -
+ +
+ +
Example 5 Use Polynomial Long Division
CHECK You can check the result of a division problem by multiplying the divisor by the quotient and adding the remainder. The result should be the dividend (what’s inside ).
+ 4x 2 – –x 2( ) x + 4( ) = x + 4( )x 2 – x x + 4( ) – 2 x + 4( ) + 4
= x 3 – – 4x + 44x 2 x 2+ – 2x – 8
= x 3 – 6x3x 2+ – 4
Synthetic division:
Is a method of dividing polynomials by an expression of the form x - k
Example 1 Using Synthetic division
Divide: x – (-4) in x – k form
-4 Coefficients of powers of x1 3 -6 -4
k
1
-4
-1
4multiply
-2
8
4
add
coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend.
ANSWER x 2 – –x 2x + 4
4+
remainder
Example 2 Using Synthetic division
Find the quotient:
3 Coefficients of powers of x2 -7 0 6 -14
k
2
6
-1
-3multiply
-3
-9
-23
add
remainder-3
-9
2 𝑥4− 7 𝑥3+0 𝑥2+6 𝑥− 14
ANSWER 𝟐 𝒙𝟑−𝒙𝟐−𝟑 𝒙−𝟑−𝟐𝟑𝒙−𝟑
Checkpoint
Find the quotient.
5x4x 3 +( )+ 1 x( + 1)÷
ANSWER
4x 4x2 +– 9 +x + 1
8–
Example 3Use Polynomial Long Division
Divide: Can’t use synthetic division because it isn’t being divided by x-k
2 𝑥2+7 𝑥− 392 𝑥−7 =
2 𝑥2 −7 𝑥- +
1 4 𝑥−39 =
𝑥
14 𝑥− 49- +
+7
1 0 remainder
𝐴𝑛𝑠𝑤𝑒𝑟 :𝑥+7+10
2 𝑥−7
Example 3 Use Polynomial Long Division
Divide:
Can’t use synthetic division because it isn’t being divided by x-k
2 𝑥4+3 𝑥3+0 𝑥2+5 𝑥− 1𝑥2−2 𝑥+2 =
2 𝑥2
2 𝑥4− 4 𝑥3+4 𝑥2- + -
+ =
+7 𝑥
+ - + -
1 0 𝑥2 −9 𝑥− 1 =
+10
1 0 𝑥2 −20 𝑥+20- + -
1 1𝑥− 21remainder
+
Homework:
WS: Dividing Polynomials