section 7.3
DESCRIPTION
Section 7.3. Products and Factors of Polynomials. Factoring the Sum and Difference of Two Cubes. Factor Theorem. (x-r) is a factor of the polynomial expression that defines the function P if and only if r is a solution of P(x) = 0, that is, if and only if P(r) = 0. In other words: - PowerPoint PPT PresentationTRANSCRIPT
Section 7.3
Products and Factors of Polynomials
Factoring the Sum and Difference of Two Cubes
Factor Theorem
• (x-r) is a factor of the polynomial expression that defines the function P if and only if r is a solution of P(x) = 0, that is, if and only if P(r) = 0.
• In other words:– Set x-r = 0– Solve for x. x = r– Plug this r in for every x in the original polynomial– Simplify– If you get 0 then (x – r) IS a factor of the polynomial
Let’s look at how to do this using the example:
4 25 4 6 ( 3)x x x x In order to use synthetic division these
two things must happen:There must be a coefficient for every possible power of the
variable.
The divisor must have a leading coefficient of 1.
#1 #2
Step #1: Write the terms of the polynomial so the degrees are in descending order.
4 3 25 0 4 6x x x x
3
Since the numerator does not contain all the powers of x,
you must include a for the .0 x
Step #2: Write the constant a of the divisor x- a to the left and write down the
coefficients.
Since the divisor , then 3 3 x a
4 3 25 0 4 6
3 5 0 4 1 6
x x x x
Step #3: Bring down the first coefficient, 5.
3 5 0 4 1 6
5
Step #5: After multiplying in the diagonals, add the column.
3 5 0 4 1 6
15
5 15
Add the column
Step #6: Multiply the sum, 15, by ; 15 3=15,
and place this number under the next coefficient,
then add the column again.
r
3 5 0 4 1 6
15 45
5 15 41
Multiply the diagonals, add the columns.
Add
41
15*3 = 45
Step #7: Repeat the same procedure as step #6.
3 5 0 4 1 6
15 45 123 372
5 15 41 12 784 3
Add Columns
Add Columns
Add Columns
Add Columns
The quotient is:
5x3 15x2 41x 124 378
x 3
Remember to place the remainder over the divisor.
Step #8: Write the quotient.
The numbers along the bottom are coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend.
Divide a polynomial by a monomial.
2
564 3 xx
2
5
2
6
2
4 3
xx
2
532 3 xx
Divide a polynomial by a monomial.
x
xxx
2
462 23
x
x
x
x
x
x
2
4
2
6
2
2 23
232 xx
Divide the following numbers.
13
5733
5733134
5253
4
5213
1 Check
13
441
5733
130
Divide the following numbers (Long Division)
11
8034
8034117
7733
3
3304
0
04 Remainder
11
4730
822 2 xxx
Dividing a polynomial by a polynomial (Long Division)
2
822
x
xxCheck
JUST WATCH THIS ONE!
Dividing a polynomial by a polynomial (Long Division)
2
822
x
xx
822 2 xxxx
xx 22 x4 8
4
84 x0
Check
)2)(4( xx
8422 xxx
822 xx
x
x2
)2(xx
x
x4 )2(4 x
JUST WATCH THIS ONE!
Dividing a polynomial by a polynomial (Long Division)
7
42133
36
x
xx Check
42137 363 xxx
JUST WATCH THIS ONE!
36x
Dividing a polynomial by a polynomial (Long Division)
7
42133
36
x
xx
3x
36 7xx 42
6
426 3 x0
Check
)6)(7( 33 xx
4276 336 xxx
4213 36 xx
)7( 33 xx
)7(6 3x
42137 363 xxx
3
6
x
x
3
36
x
x
JUST WATCH THIS ONE!
Dividing a polynomial by a polynomial (Long Division)
1
124
x
xx
1001 234 xxxxx
NOW YOU WRITE THIS ONE
Dividing a polynomial by a polynomial (Long Division)
1
124
x
xx
1001 234 xxxxx
3x
34 xx 3x 2x
2x
23 xx 22x x0
x2 x2
xx 22 2 x2 1
2
12 x22 x3
1
3
x
Dividing a polynomial by a polynomial (Long Division)
yy
yyy
523
19730152
23
7301915253 232 yyyyy
NOW YOU WRITE THIS ONE
Dividing a polynomial by a polynomial (Long Division)
yy
yyy
523
19730152
23
7301915253 232 yyyyyy5
yyy 102515 23 26y y20
2
4106 2 yy11y10
7
253
11102
yy
y
Homework Problems
1.
2.
3.
4.