6.5 Warm Up
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Transcript of 6.5 Warm Up
6.5 Warm Up
1. Factor 8x3 + 125.
2. Factor 5x3 + 10x2 – x – 2.
3. Factor 200x6 – 2x4.
4. Find the product of (2x – 3)(2x – 5).
5. Find the product of (5x + 2y)3
6.5 Polynomial Division
Long division( ) ( )
( )( ) ( )
n x r xq x
d x d x
Numerator
Denominator Quotient Remainder expressed as a fraction.
Long Division Example( ) ( )
( )( ) ( )
n x r xq x
d x d x
4 3 3 2Divide (2 2 10 9) by ( 5)x x x x x
3 2 4 3
4
3
2
5 2 2 10 9
22 - (2
x
x x x x x
xx x
x
4 32 10 )
---------------------
0 9
x x
Remainder
3 2
9
5x x
Long Division Example 2 3 2 2Divide using long division. 2 2 3 1x x x x
Notice x2 –1 is missing a term.
2 3 2
3
2
0 1 2 2 3x x x x x
xx
x
2 3 2
3
2
0 1 2 2 3
x
x x x x x
xx
x
3 2
2
2
2
-( 0 )
_____________
2 3
22
x x x
x x
x
x
2 3 2
3
2
2
0 1 2 2 3
x
x x x x x
xx
x
3 2
2
2
2
-( 0 )
_____________
2 3 3
22
x x x
x x
x
x
2 - (2 0 2)
____________
x x
3 5x Remainder
2
3 5
1
x
x
Classwork
Textbook page 356 probs 4 – 7.
Synthetic Division
Synthetic division can be used to divide polynomials by an expression in the form of x - k.
Example: Divide (x3 – 8x + 3) by (x + 3).
x + 3 is in the form of x – k.
x + 3 = 0
x = -3 -3 1 0 -8 3
1
-3
-3
9
1
-3
0 Remainder
x2 - 3x + 1Quotient
Classwork
Textbook page 356 probs 8 - 11.
Remainder and Factor Theorem
Remainder Theorem
If a polynomial f(x) is divided by x – k, then the remainder is r = f(k).
Factor Theorem
A polynomial f(x) has a factor x – k if and only if f(k) = 0.
Example: Problem 7 page 116 had a remainder of –7. 2
2
If we evaluate ( ) +2 15 when 4
( 4) ( 4) 2( 4) 15 7
f x x x x
f
f(-4) = the remainder.
Synthetic Division ContinuedExample: Factor (x3 – 8x + 3) given (x + 3) is a factor.
x2 - 3x + 1
1 0 -8 3
1
-3
-3
9
1
-3
0
-3
Continue factoring
( 3) 9 4(1)(1) 3 5
2(1) 2x
So, 3 3 5 3 5
8 3 ( 3)( )( )2 2
x x x x x
Synthetic Division Example 2Given one zero of the polynomial function, find the other zeros.
3 2( ) 2 14 56 40, 10f x x x x x
10 2 -14 –56 -40
220
660
4400
2x2 + 6x + 4
(2x + 4)(x + 1)
(2x + 4) = 0 (x + 1) = 0
x = -2 x = -1
So, the zeros of the polynomial are 10, -1, and –2.
A Geometric Interpretation
3 2( ) 2 14 56 40, 10f x x x x x
-18-15-12 -9 -6 -3 3 6 9 12 15 18
-3920-3430-2940-2450-1960-1470-980-490
490980
-4 -2 2 4 6 8 10 12 14
-10
-8
-6
-4
-2
2
4
6
8
10
The real zeros are the x-intercepts of the graph.
A Geometry ProblemGiven the expression for the volume of a rectangular prism, find an expression for the missing dimension.
x + 5
x + 1
?
3 23 8 45 50V x x x
x = -1 -1 3 8 -45 -50
3-3 -55 -50
500
x = -5 -5 3 5 -50
3-15-10
500
3x -10
Classwork
Textbook page 356 probs 12 - 13.
Homework Textbook page 356 probs 15 – 45 1st col.