ACTIVITY 31: Dividing Polynomials (Section 4.2, pp. 325-331)
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ACTIVITY 31:
Dividing Polynomials (Section 4.2, pp. 325-331)
Example 1:
Divide 63 by 12.
rq 12*63 122436486072
312*563
Division Algorithm:
If P(x) and D(x) are polynomials, with D(x) ≠ 0, then there exist unique polynomials Q(x) and R(x), where R(x) is either 0 or of degree strictly less than the degree of D(x), such that
P(x) = Q(x)D(x) + R(x)The polynomials P(x) and D(x) are called the dividend and divisor, respectively; Q(x) is the quotientand R(x) is the remainder.
Example 2:
Divide the polynomial P(x) = 2x2 − x − 4 by D(x) = x − 3.
2x2 − x − 4 x − 3
2x
2x2 – 6x-2x2 + 6x
5x – 4
+ 5
5x – 15 -5x + 15
11
)(xQ
)(xR
)()()()( xRxDxQxP 1135242 2 xxxx
Example 3:
x4 − x3 + 4x + 2x2 + 3
x2
x4 + 3x2 –x4 – 3x2
− x3 – 3x2 + 4x + 2
– x
– x3 – 3x + x3 + 3x
– 3x2 + 7x + 2
)(xQ
)(xR
)()()()( xRxDxQxP 1173324 2234 xxxxxxx
Divide the polynomial P(x) = x4 − x3 + 4x + 2 by D(x) = x2 + 3.
– 3
– 3x2 – 9 +3x2 + 9
7x + 11
Synthetic Division:
Use synthetic division to divide the polynomialP(x) = 2x2 − x − 4 by D(x) = x − 3.
root 23 1 4
2
6
5
15
11
)(xQ )(xR
)()()()( xRxDxQxP 1135242 2 xxxx
52)( xxQ
Example 4:
Use synthetic division to find the quotient Q(x) and the remainder R(x) when:
f(x) = 3x3 + 2x2 − x + 3 is divided by g(x) = x − 4.
34 2 1
3
12
14
56
55
)(xQ )(xR55143)( 2 xxxQ
3
220
223
223)( xR
Example 5:
Use synthetic division to find the quotient Q(x) and the remainder R(x) when:
f(x) = x5 − 4x3 + x is divided by g(x) = x + 3.
13 0 4
1
3
3
9
5
)(xQ )(xR461553)( 234 xxxxxQ
0
15
15
138)( xR
1 0
45
46
138
138
Remainder Theorem:
If the polynomial P(x) is divided by D(x) = x − c, then
)()()()( xRxDxQxP
Plugging in x=c to the above equation one sees that
becomes
)())(()( xRcxxQxP
)())(()( cRcccQcP )()( cRcP
Example 6:
Let P(x) = x3 + 2x2 − 7.(a) Find the quotient and the remainder when P(x) is divided by x + 2.(b) Use the Remainder Theorem to find P(−2).
12 2 0
1
2
0
0
0
)(xQ )(xR
2)( xxQ
7
0
77)( xR
)()()()( xRxDxQxP 7272 223 xxxx
)2(P 7
Factor Theorem:
The number c is a zero of P(x) if and only if x−c is a factor of P(x); that is, P(x) = Q(x) · (x − c) for some polynomial Q(x). In other words, in Synthetic division the R(x) = 0 that is the last term is zero.
Example 7:
Use the Factor Theorem to determine whether x + 2 is a factor of f(x) = 3x6 + 2x3 − 176.
32 0 0
3
6
6
12
12
2
24
22
0 0
44
44
88
88
176176
0YES!!!!!!!!!!
Example 8:
Find a polynomial of degree 3 that has zeros 1, −2, and 3, and in which the coefficient of x2 is 3.
1x 2x 3xa
3222 xxxxa
322 xxxa 6233 223 xxxxxa
652 23 xxxa
aaxaxax 652 23
32 a
2
3
a
1x 2x 3x2
3
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