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Lesson 3.4 Zeros of Polynomials
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Rational Zeros Theorem
List all possible rational zeros given by the Rational
Zeros Theorem.
1. 4 33 6 8P x x x x
Video tutorial
2. 5 312 6 2 8P x x x x
Video tutorial
Find all rational zeros of the polynomial, and write the
polynomial in factored form.
If the polynomial 11 1 0( ) ...n nn nP x a x a x a x a has
integer coefficients, thenevery rational zero ofP is of the
form
p
q where p is a factor of the constant
coefficient0
a and q is a factor of the leading coefficientn
a
http://screencast.com/t/pMJE8WEaMhttp://screencast.com/t/pMJE8WEaMhttp://screencast.com/t/VGOuiE5QghZhttp://screencast.com/t/VGOuiE5QghZhttp://screencast.com/t/VGOuiE5QghZhttp://screencast.com/t/pMJE8WEaM
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Lesson 3.4 Zeros of Polynomials
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3. 3 27 14 8P x x x x
Video tutorial
http://screencast.com/t/eBDbEqrQhttp://screencast.com/t/eBDbEqrQhttp://screencast.com/t/eBDbEqrQ
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7/27/2019 Lesson 3.4 Video Tutorials Zeros of
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Lesson 3.4 Zeros of Polynomials
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Find all rational zeros of the polynomial, and write the
polynomial in factored form.
4. 4 3 2
2 3 8 4P x x x x x
Video tutorial
http://screencast.com/t/PCgbxe0hhttp://screencast.com/t/PCgbxe0h
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Lesson 3.4 Zeros of Polynomials
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Find all rational zeros of the polynomial, and write the
polynomial in factored form.
5. 4 3 2
2 19 9 9P x x x x x
Video tutorial
http://screencast.com/t/9VY4g1XuBvhttp://screencast.com/t/9VY4g1XuBvhttp://screencast.com/t/9VY4g1XuBv
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Lesson 3.4 Zeros of Polynomials
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Find all real zeros of the polynomial. Use the quadratic
formula if necessary.
6. 3 2
5 2 12P x x x x
Video tutorial
http://screencast.com/t/YfK2Ft1pehttp://screencast.com/t/YfK2Ft1pe
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Lesson 3.4 Zeros of Polynomials
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Find all real zeros of the polynomial. Use the quadratic
formula if necessary.
7. 4 3 2
2 2 3 2P x x x x x Video tutorial
http://screencast.com/t/rppcb8VmEhttp://screencast.com/t/rppcb8VmE
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Lesson 3.4 Zeros of Polynomials
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A. The Fundamental Theorem of Algebra
A polynomial Pis given. (a) Find all zeros ofP, real and
complex. (b) Factor Pcompletely. (c) State the
multiplicity of each zero.
1. 5 39P x x x
Video tutorial
2. 4 26 9P x x x
Video tutorial
If f(x) is a polynomial of degree n, where n> 0, then fhas at
least one zero inthe complex number system.
This means that if the degree is 3, you will have 3 zeros (2
could be imaginaryand 1 could be real, or you could have 3 real
zeros)
Conjugate Zeros Theorem: Complex zeros always come in conjugate
pairs (a + bi) and (abi) 2iand -2i, 5 + iand 5i , 3 + 2iand 32i
http://screencast.com/t/QDrnZ7txhttp://screencast.com/t/QDrnZ7txhttp://screencast.com/t/0Acp8dOp8Rhttp://screencast.com/t/0Acp8dOp8Rhttp://screencast.com/t/QDrnZ7tx
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Lesson 3.4 Zeros of Polynomials
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A polynomial Pis given. (a) Find all zeros ofP, real and
complex. (b) Factor Pcompletely. (c) State the
multiplicity of each zero.
3. 3
8P x x
Video tutorial
4.
2 8 17P x x x
Video tutorial
http://screencast.com/t/AXKijz4pfvhttp://screencast.com/t/AXKijz4pfvhttp://screencast.com/t/F6b2lAwiO6http://screencast.com/t/F6b2lAwiO6http://screencast.com/t/AXKijz4pfv
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Lesson 3.4 Zeros of Polynomials
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A polynomial Pis given. (a) Find all zeros ofP, real and
complex. (b) Factor Pcompletely. (c) State the
multiplicity of each zero.
5. 4
625P x x
Video tutorial
6. 3 2
P x x x x
Video tutorial
http://screencast.com/t/ni58HzLUghttp://screencast.com/t/ni58HzLUghttp://screencast.com/t/TIpmpXvaYhttp://screencast.com/t/TIpmpXvaYhttp://screencast.com/t/ni58HzLUg
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Lesson 3.4 Zeros of Polynomials
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A polynomial Pis given. (a) Find all zeros ofP, real and
complex. (b) Factor Pcompletely. (c) State the
multiplicity of each zero.
7. 5 3
7P x x x
Video tutorial
8. 6 316 64P x x x
Video tutorial
http://screencast.com/t/3bbyLW7tCt8http://screencast.com/t/3bbyLW7tCt8http://screencast.com/t/LWe5JvMrQhttp://screencast.com/t/LWe5JvMrQhttp://screencast.com/t/3bbyLW7tCt8
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Lesson 3.4 Zeros of Polynomials
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Find a polynomial with integer coefficients that satisfies
the given conditions.
1. P has degree 2 and zeros 1 2i and 1 2i
Video tutorial
2. P has degree 3 and zeros 0 and i
Video tutorial
http://screencast.com/t/6yb2tFKmLryIhttp://screencast.com/t/6yb2tFKmLryIhttp://screencast.com/t/aNWdWjz5iH2http://screencast.com/t/aNWdWjz5iH2http://screencast.com/t/aNWdWjz5iH2http://screencast.com/t/6yb2tFKmLryI
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7/27/2019 Lesson 3.4 Video Tutorials Zeros of
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Lesson 3.4 Zeros of Polynomials
12
A. The Fundamental Theorem of Algebra
A polynomial Pis given. (a) Find all zeros ofP, real and
complex. (b) Factor Pcompletely. (c) State the
multiplicity of each zero.
1. 5 39P x x x
Video tutorial
2. 4 26 9P x x x
Video tutorial
If f(x) is a polynomial of degree n, where n> 0, then fhas at
least one zero inthe complex number system.
This means that if the degree is 3, you will have 3 zeros (2
could be imaginaryand 1 could be real, or you could have 3 real
zeros)
Conjugate Zeros Theorem: Complex zeros always come in conjugate
pairs (a + bi) and (abi) 2iand -2i, 5 + iand 5i , 3 + 2iand 32i
http://screencast.com/t/QDrnZ7txhttp://screencast.com/t/QDrnZ7txhttp://screencast.com/t/0Acp8dOp8Rhttp://screencast.com/t/0Acp8dOp8Rhttp://screencast.com/t/QDrnZ7tx
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7/27/2019 Lesson 3.4 Video Tutorials Zeros of
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Lesson 3.4 Zeros of Polynomials
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A polynomial Pis given. (a) Find all zeros ofP, real and
complex. (b) Factor Pcompletely. (c) State the
multiplicity of each zero.
3. 3
8P x x
Video tutorial
4.
2 8 17P x x x
Video tutorial
http://screencast.com/t/AXKijz4pfvhttp://screencast.com/t/AXKijz4pfvhttp://screencast.com/t/F6b2lAwiO6http://screencast.com/t/F6b2lAwiO6http://screencast.com/t/AXKijz4pfv
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Lesson 3.4 Zeros of Polynomials
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A polynomial Pis given. (a) Find all zeros ofP, real and
complex. (b) Factor Pcompletely. (c) State the
multiplicity of each zero.
5. 4
625P x x
Video tutorial
6. 3 2
P x x x x
Video tutorial
http://screencast.com/t/ni58HzLUghttp://screencast.com/t/ni58HzLUghttp://screencast.com/t/TIpmpXvaYhttp://screencast.com/t/TIpmpXvaYhttp://screencast.com/t/ni58HzLUg
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Lesson 3.4 Zeros of Polynomials
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A polynomial Pis given. (a) Find all zeros ofP, real and
complex. (b) Factor Pcompletely. (c) State the
multiplicity of each zero.
7. 5 3
7P x x x
Video tutorial
8. 6 316 64P x x x
Video tutorial
http://screencast.com/t/3bbyLW7tCt8http://screencast.com/t/3bbyLW7tCt8http://screencast.com/t/LWe5JvMrQhttp://screencast.com/t/LWe5JvMrQhttp://screencast.com/t/3bbyLW7tCt8
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Lesson 3.4 Zeros of Polynomials
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Find a polynomial with integer coefficients that satisfies
the given conditions.
1. P has degree 2 and zeros 1 2i and 1 2i
Video tutorial
2. P has degree 3 and zeros 0 and i
Video tutorial
http://screencast.com/t/6yb2tFKmLryIhttp://screencast.com/t/6yb2tFKmLryIhttp://screencast.com/t/aNWdWjz5iH2http://screencast.com/t/aNWdWjz5iH2http://screencast.com/t/aNWdWjz5iH2http://screencast.com/t/6yb2tFKmLryI
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7/27/2019 Lesson 3.4 Video Tutorials Zeros of
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Lesson 3.4 Zeros of Polynomials
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Find a polynomial with integer coefficients that satisfies
the given conditions.
1. P has degree 4 and zeros 1 2i
and 1, with 1 a zerowith multiplicity 2.
Video tutorial
http://screencast.com/t/CoxyGDZcuihttp://screencast.com/t/CoxyGDZcuihttp://screencast.com/t/CoxyGDZcui
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Lesson 3.4 Zeros of Polynomials
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Find all zeros of the polynomial.
1. 3 27 17 15P x x x x
Video tutorial
http://www.screencast.com/t/D5Ml1AwL1http://www.screencast.com/t/D5Ml1AwL1http://www.screencast.com/t/D5Ml1AwL1
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Lesson 3.4 Zeros of Polynomials
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Find all real zeros of the polynomial.
2. 3 27 18 18P x x x x
Video tutorial
http://www.screencast.com/t/L5YEdhxyqcihttp://www.screencast.com/t/L5YEdhxyqcihttp://www.screencast.com/t/L5YEdhxyqci
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Lesson 3.4 Zeros of Polynomials
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Find all real zeros of the polynomial.
3. 3 6P x x x
Video tutorial
http://www.screencast.com/t/D8gApoMn4http://www.screencast.com/t/D8gApoMn4
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Lesson 3.4 Zeros of Polynomials
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Find all real zeros of the polynomial.
4. 4 3 22 2 2 3P x x x x x
Video tutorial
http://screencast.com/t/HEHZGdnArx2http://screencast.com/t/HEHZGdnArx2
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Lesson 3.4 Zeros of Polynomials
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A Polynomial Pis given. (a) Factor Pinto linear and
irreducible quadratic factors with real coefficients. (b)
Factor Pcompletely into linear factors with complex
coefficients.
1. 3 2 4P x x x
Video
http://www.screencast.com/t/QxeRV59hd471http://www.screencast.com/t/QxeRV59hd471
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7/27/2019 Lesson 3.4 Video Tutorials Zeros of
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Lesson 3.4 Zeros of Polynomials
A Polynomial Pis given. (a) Factor Pinto linear and
irreducible quadratic factors with real coefficients. (b)
Factor Pcompletely into linear factors with complex
coefficients.
2. 4 28 16P x x x
Video
http://www.screencast.com/t/ET4NUItOg8DRhttp://www.screencast.com/t/ET4NUItOg8DRhttp://www.screencast.com/t/ET4NUItOg8DR
