warm-up 2/24
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Warm-Up 2/241.
B
12
6
6
Rigor:You will learn how to divide polynomials and use
the Remainder and Factor Theorems.
Relevance:You will be able to use graphs and equations of
polynomial functions to solve real world problems.
2-3 The Remainder and Factor Theorems
−3 𝑥+9
Example 1: Use long division to factor polynomial.6 𝑥3−25 𝑥2+18 𝑥+9 ; (𝑥−3 )
6 𝑥3−25 𝑥2+18 𝑥+9𝑥−36 𝑥2
6 𝑥3−18𝑥2−6 𝑥3+18 𝑥2
−7 𝑥2+18 𝑥+9
−7 𝑥
−7 𝑥2+21𝑥+7 𝑥2−21𝑥
−3
−3 𝑥+9+3 𝑥−90
(𝑥−3 )(6 𝑥2−7𝑥−3)(𝑥−3 )(2 𝑥−3)(3𝑥+1)So there are real zeros at x = 3, , and .
3 𝑥−3
Example 2: Divide the polynomial.9 𝑥3−𝑥−3 ; (3 𝑥+2 )
9 𝑥3+0 𝑥2−𝑥−33 𝑥+23 𝑥2
9 𝑥3+6 𝑥2−9𝑥3−6 𝑥2
−6 𝑥2−𝑥−3
−2 𝑥
−6 𝑥2−4 𝑥+6 𝑥2+4 𝑥
+1
3 𝑥+2−3 𝑥−2−5
9𝑥3−𝑥−33 𝑥+2
=3 𝑥2−2𝑥+1+−5
3𝑥+2,𝑥 ≠− 2
39𝑥3−𝑥−3
3 𝑥+2=3 𝑥2−2𝑥+1− 5
3𝑥+2,𝑥≠− 2
3
𝑥−4
Example 3: Divide the polynomial.2 𝑥4−4 𝑥3+13 𝑥2+3 𝑥−11 ; (𝑥2−2𝑥+7 )
2 𝑥4−4 𝑥3+13 𝑥2+3 𝑥−11𝑥2−2 𝑥+72 𝑥2
2 𝑥4−4 𝑥3+14 𝑥2−2 𝑥4+4 𝑥3−14 𝑥2
−𝑥2+3 𝑥−11
−1
−𝑥2+2𝑥−7+𝑥2−2𝑥+7
2𝑥4−4 𝑥3+13 𝑥2+3 𝑥−11𝑥2−2 𝑥+7
=2 𝑥2−1+𝑥−4
𝑥2−2 𝑥+7
Example 4a: Divide the polynomial using synthetic division.(2 𝑥4−5 𝑥2+5 𝑥−2)÷ (𝑥+2 )
– 5
– 6
– 4
2 0 5
↓
– 2
– 4 8
32 – 1
2
0
– 2
2𝑥4−5 𝑥2+5𝑥−2𝑥+2
=2𝑥3−4 𝑥2+3 𝑥−1
2 𝑥3−4 𝑥2+3 𝑥−1
Example 4b: Divide the polynomial using synthetic division.(10 𝑥3−13𝑥2+5 𝑥−14 )÷ (2 𝑥−3 )
52
6
1
5 − 132 – 7
↓ 152
32
45 – 1
32
5 𝑥2+𝑥+4− 1
𝑥− 32
=5 𝑥2+𝑥+4− 22𝑥−3
(10 𝑥3−13𝑥2+5 𝑥−14 )(2 𝑥−3)
(10 𝑥3−13𝑥2+5 𝑥−14 )÷2(2 𝑥−3)÷2
=5 𝑥3− 13
2 𝑥2+52 𝑥−7
𝑥− 32
Example 6a: Use the Factor Theorem to determine if the binomials are factors of f(x). Write f(x) in factor form if possible.𝑓 (𝑥 )=4 𝑥4+21𝑥3+25 𝑥2−5𝑥+3 ; (𝑥−1) , (𝑥+3 )
25
50
25
4 21 – 5
↓
3
4 25
504 45
45
48
1
, so is not a factor.
25
6
9
4 21 – 5
↓
3
– 12 – 27
– 2 4 1
– 3
0
– 3
, so is a factor.
𝑓 (𝑥 )=(𝑥+3 )(4 𝑥3+9𝑥2−2𝑥+1)
Example 6b: Use the Factor Theorem to determine if the binomials are factors of f(x). Write f(x) in factor form if possible.𝑓 (𝑥 )=2𝑥3−𝑥2−41𝑥−20 ;(𝑥+4) , (𝑥−5 )
– 41
20
– 9
2 – 1 – 20
↓ – 8 36
– 5 2 0
– 4
, so is a factor.
𝑓 (𝑥 )=(𝑥+4 )(𝑥−5)(2 𝑥+1)
– 5
1
2 – 9
↓ 10 5
02
5
, so is a factor.
√−1math!
2-3 Assignment: TX p115, 4-44 EOE
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