9-4 notes
TRANSCRIPT
SECTION 9-4Division and the Remainder Theorem
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WARM-UP1. Divide. Check your answer by substituting 5 for x.
12x4
3x3
2. Divide. Check your answer by substituting 2 for x.
12x4 −6x3
3x3
WARM-UP1. Divide. Check your answer by substituting 5 for x.
12x4
3x3
2. Divide. Check your answer by substituting 2 for x.
12x4 −6x3
3x3
= 4x
WARM-UP1. Divide. Check your answer by substituting 5 for x.
12x4
3x3
2. Divide. Check your answer by substituting 2 for x.
12x4 −6x3
3x3
= 4x
12i54
3i53= 12i625
3i125 = 7500375 = 20 = 4i5
WARM-UP1. Divide. Check your answer by substituting 5 for x.
12x4
3x3
2. Divide. Check your answer by substituting 2 for x.
12x4 −6x3
3x3
= 4x
= 4x − 2
12i54
3i53= 12i625
3i125 = 7500375 = 20 = 4i5
WARM-UP1. Divide. Check your answer by substituting 5 for x.
12x4
3x3
2. Divide. Check your answer by substituting 2 for x.
12x4 −6x3
3x3
= 4x
= 4x − 2
12i54
3i53= 12i625
3i125 = 7500375 = 20 = 4i5
12i24 −6i23
3i23= 192−48
24 = 14424 = 6 = 4i2 − 2
WARM-UP
3. Find f(a).
a3 − 8a2 +19a −12 = (a − 3) f (a)
WARM-UP
3. Find f(a).
a3 − 8a2 +19a −12 = (a − 3) f (a)
f (a) = a2 − 5a + 4
EXAMPLE 1−2x3 −7x2 +10x−25
x+5
EXAMPLE 1−2x3 −7x2 +10x−25
x+5
x + 5 −2x3 − 7x2 +10x − 25
EXAMPLE 1−2x3 −7x2 +10x−25
x+5
x + 5 −2x3 − 7x2 +10x − 25−2x2
EXAMPLE 1−2x3 −7x2 +10x−25
x+5
x + 5 −2x3 − 7x2 +10x − 25−2x2
−2x3 −10x2
EXAMPLE 1−2x3 −7x2 +10x−25
x+5
x + 5 −2x3 − 7x2 +10x − 25−2x2
−2x3 −10x2
3x2 +10x − 25
EXAMPLE 1−2x3 −7x2 +10x−25
x+5
x + 5 −2x3 − 7x2 +10x − 25−2x2
−2x3 −10x2
3x2 +10x − 25
+3x
EXAMPLE 1−2x3 −7x2 +10x−25
x+5
x + 5 −2x3 − 7x2 +10x − 25−2x2
−2x3 −10x2
3x2 +10x − 25
+3x
3x2 +15x
EXAMPLE 1−2x3 −7x2 +10x−25
x+5
x + 5 −2x3 − 7x2 +10x − 25−2x2
−2x3 −10x2
3x2 +10x − 25
+3x
3x2 +15x−5x − 25
EXAMPLE 1−2x3 −7x2 +10x−25
x+5
x + 5 −2x3 − 7x2 +10x − 25−2x2
−2x3 −10x2
3x2 +10x − 25
+3x
3x2 +15x−5x − 25
−5
EXAMPLE 1−2x3 −7x2 +10x−25
x+5
x + 5 −2x3 − 7x2 +10x − 25−2x2
−2x3 −10x2
3x2 +10x − 25
+3x
3x2 +15x−5x − 25
−5
−5x − 25
EXAMPLE 1−2x3 −7x2 +10x−25
x+5
x + 5 −2x3 − 7x2 +10x − 25−2x2
−2x3 −10x2
3x2 +10x − 25
+3x
3x2 +15x−5x − 25
−5
−5x − 250
EXAMPLE 1−2x3 −7x2 +10x−25
x+5
x + 5 −2x3 − 7x2 +10x − 25−2x2
−2x3 −10x2
3x2 +10x − 25
+3x
3x2 +15x−5x − 25
−5
−5x − 250
−2x2 + 3x − 5
EXAMPLE 210y4 +7y2 −2y+3
2y2 +3y
EXAMPLE 210y4 +7y2 −2y+3
2y2 +3y
2y2 + 3y 10y4 + 0y3 + 7y2 − 2y + 3
EXAMPLE 210y4 +7y2 −2y+3
2y2 +3y
2y2 + 3y 10y4 + 0y3 + 7y2 − 2y + 35y2
EXAMPLE 210y4 +7y2 −2y+3
2y2 +3y
2y2 + 3y 10y4 + 0y3 + 7y2 − 2y + 35y2
10y4 +15y3
EXAMPLE 210y4 +7y2 −2y+3
2y2 +3y
2y2 + 3y 10y4 + 0y3 + 7y2 − 2y + 35y2
10y4 +15y3
−15y3 + 7y2
EXAMPLE 210y4 +7y2 −2y+3
2y2 +3y
2y2 + 3y 10y4 + 0y3 + 7y2 − 2y + 35y2
10y4 +15y3
−15y3 + 7y2
− 152 y
EXAMPLE 210y4 +7y2 −2y+3
2y2 +3y
2y2 + 3y 10y4 + 0y3 + 7y2 − 2y + 35y2
10y4 +15y3
−15y3 + 7y2
− 152 y
−15y3 − 452 y
2
EXAMPLE 210y4 +7y2 −2y+3
2y2 +3y
2y2 + 3y 10y4 + 0y3 + 7y2 − 2y + 35y2
10y4 +15y3
−15y3 + 7y2
− 152 y
−15y3 − 452 y
2
592 y
2 − 2y
EXAMPLE 210y4 +7y2 −2y+3
2y2 +3y
2y2 + 3y 10y4 + 0y3 + 7y2 − 2y + 35y2
10y4 +15y3
−15y3 + 7y2
− 152 y
−15y3 − 452 y
2
592 y
2 − 2y
+ 594
EXAMPLE 210y4 +7y2 −2y+3
2y2 +3y
2y2 + 3y 10y4 + 0y3 + 7y2 − 2y + 35y2
10y4 +15y3
−15y3 + 7y2
− 152 y
−15y3 − 452 y
2
592 y
2 − 2y
+ 594
592 y
2 + 1774 y
EXAMPLE 210y4 +7y2 −2y+3
2y2 +3y
2y2 + 3y 10y4 + 0y3 + 7y2 − 2y + 35y2
10y4 +15y3
−15y3 + 7y2
− 152 y
−15y3 − 452 y
2
592 y
2 − 2y
+ 594
592 y
2 + 1774 y
− 1854 y + 3
EXAMPLE 210y4 +7y2 −2y+3
2y2 +3y
2y2 + 3y 10y4 + 0y3 + 7y2 − 2y + 35y2
10y4 +15y3
−15y3 + 7y2
− 152 y
−15y3 − 452 y
2
592 y
2 − 2y
+ 594
592 y
2 + 1774 y
− 1854 y + 3
5y2 − 152 y +
594
R :− 1854 y + 3
REMAINDER THEOREM
REMAINDER THEOREM
If a polynomial f(x) is divided by x - c, then the remainder is f(c).
EXAMPLE 3Divide and check your answer using the Remainder Thm.
EXAMPLE 3Divide and check your answer using the Remainder Thm.
x + 4 6x3 + 3x2 +14
EXAMPLE 3Divide and check your answer using the Remainder Thm.
x + 4 6x3 + 3x2 +146x2
EXAMPLE 3Divide and check your answer using the Remainder Thm.
x + 4 6x3 + 3x2 +146x2
6x3 + 24x2
EXAMPLE 3Divide and check your answer using the Remainder Thm.
x + 4 6x3 + 3x2 +146x2
6x3 + 24x2
−21x2 + 0x
EXAMPLE 3Divide and check your answer using the Remainder Thm.
x + 4 6x3 + 3x2 +146x2
6x3 + 24x2
−21x2 + 0x
−21x
EXAMPLE 3Divide and check your answer using the Remainder Thm.
x + 4 6x3 + 3x2 +146x2
6x3 + 24x2
−21x2 + 0x
−21x
−21x2 − 84x
EXAMPLE 3Divide and check your answer using the Remainder Thm.
x + 4 6x3 + 3x2 +146x2
6x3 + 24x2
−21x2 + 0x
−21x
−21x2 − 84x84x +14
EXAMPLE 3Divide and check your answer using the Remainder Thm.
x + 4 6x3 + 3x2 +146x2
6x3 + 24x2
−21x2 + 0x
−21x
−21x2 − 84x84x +14
+84
EXAMPLE 3Divide and check your answer using the Remainder Thm.
x + 4 6x3 + 3x2 +146x2
6x3 + 24x2
−21x2 + 0x
−21x
−21x2 − 84x84x +14
+84
84x + 336
EXAMPLE 3Divide and check your answer using the Remainder Thm.
x + 4 6x3 + 3x2 +146x2
6x3 + 24x2
−21x2 + 0x
−21x
−21x2 − 84x84x +14
+84
84x + 336−322
EXAMPLE 3Divide and check your answer using the Remainder Thm.
x + 4 6x3 + 3x2 +146x2
6x3 + 24x2
−21x2 + 0x
−21x
−21x2 − 84x84x +14
+84
84x + 336−322
6x2 − 21x + 84,R :−322
EXAMPLE 3Divide and check your answer using the Remainder Thm.
x + 4 6x3 + 3x2 +146x2
6x3 + 24x2
−21x2 + 0x
−21x
−21x2 − 84x84x +14
+84
84x + 336−322
6x2 − 21x + 84,R :−322d(−4) = 6(−4)3 + 3(−4)2 +14
EXAMPLE 3Divide and check your answer using the Remainder Thm.
x + 4 6x3 + 3x2 +146x2
6x3 + 24x2
−21x2 + 0x
−21x
−21x2 − 84x84x +14
+84
84x + 336−322
6x2 − 21x + 84,R :−322d(−4) = 6(−4)3 + 3(−4)2 +14
= −384 + 48 +14
EXAMPLE 3Divide and check your answer using the Remainder Thm.
x + 4 6x3 + 3x2 +146x2
6x3 + 24x2
−21x2 + 0x
−21x
−21x2 − 84x84x +14
+84
84x + 336−322
6x2 − 21x + 84,R :−322d(−4) = 6(−4)3 + 3(−4)2 +14
= −384 + 48 +14= −322
HOMEWORK
HOMEWORK
p. 581 #1-15