lesson 4-1 polynomial functions. vocabulary polynomial in one variable – an expression of the...
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![Page 1: Lesson 4-1 Polynomial Functions. Vocabulary Polynomial in one variable – an expression of the form. The coefficients represent complex numbers (real or](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649eb75503460f94bc0a29/html5/thumbnails/1.jpg)
Lesson 4-1
Polynomial Functions
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Vocabulary
• Polynomial in one variable – an expression of the form . The coefficients represent complex numbers (real or imaginary) is not zero, and n represents a nonnegative integer.
• Degree – the greatest exponent of a polynomials variable.
• Leading coefficient – the coefficient of the variable with the greatest exponent.
• Polynomial function – a function where P(x) is a polynomial in one variable.
![Page 3: Lesson 4-1 Polynomial Functions. Vocabulary Polynomial in one variable – an expression of the form. The coefficients represent complex numbers (real or](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649eb75503460f94bc0a29/html5/thumbnails/3.jpg)
• Polynomial equation - a polynomial that is set equal to zero.• Root – a solution of the equation P(x) = 0• Imaginary Number – a complex number of the form where
and is the imaginary number.• Complex numbers – the imaginary numbers combined with
the real numbers• Pure Imaginary Number – the complex number when a = 0
and • Fundamental Theorem of Algebra – Every polynomial
equation with degree greater than zero has at least one root in the set of complex numbers
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Polynomial Functions
![Page 5: Lesson 4-1 Polynomial Functions. Vocabulary Polynomial in one variable – an expression of the form. The coefficients represent complex numbers (real or](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649eb75503460f94bc0a29/html5/thumbnails/5.jpg)
State the degree and the leading coefficient of the polynomial
13)( 234 xxxxf
Degree of 4 and leading coefficient of 3Determine whether -2 is a zero of f(x) f(-2) = 59 Not a zero
![Page 6: Lesson 4-1 Polynomial Functions. Vocabulary Polynomial in one variable – an expression of the form. The coefficients represent complex numbers (real or](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649eb75503460f94bc0a29/html5/thumbnails/6.jpg)
State the degree and the leading coefficient of the polynomial
1835)( 23 xxxxf
Degree of 3 and leading coefficient of 1Determine whether 6 is a zero of f(x) f(6) = 0 Yes it is a zero
![Page 7: Lesson 4-1 Polynomial Functions. Vocabulary Polynomial in one variable – an expression of the form. The coefficients represent complex numbers (real or](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649eb75503460f94bc0a29/html5/thumbnails/7.jpg)
State the degree and the leading coefficient of the polynomial
285)( 52 xxxf
Degree of 5 and leading coefficient of 8Determine whether 1 is a zero of f(x) f(1) = 15 No
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Example 2
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Fundamental Theorem of Algebra
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Corollary to the Fundamental Theorem of Algebra
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![Page 12: Lesson 4-1 Polynomial Functions. Vocabulary Polynomial in one variable – an expression of the form. The coefficients represent complex numbers (real or](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649eb75503460f94bc0a29/html5/thumbnails/12.jpg)
![Page 13: Lesson 4-1 Polynomial Functions. Vocabulary Polynomial in one variable – an expression of the form. The coefficients represent complex numbers (real or](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649eb75503460f94bc0a29/html5/thumbnails/13.jpg)
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Example 3
![Page 15: Lesson 4-1 Polynomial Functions. Vocabulary Polynomial in one variable – an expression of the form. The coefficients represent complex numbers (real or](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649eb75503460f94bc0a29/html5/thumbnails/15.jpg)
Write a polynomial equation of least degree with roots 2, 3i, and -3i
01892
0)9)(2(
0)9)(2(
0)3)(3)(2(
23
2
22
xxx
xx
ixx
ixixx
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Write a polynomial equation of least degree with roots -5 and 7
0352
0)7)(5(2
xx
xx
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Write a polynomial equation of least degree with roots 6, 2i, -2i, i,-i
02443056
0)45)(6(
0)1)(4)(6(
0))(4)(6(
0))()(2)(2)(6(
2345
24
22
2222
xxxxx
xxx
xxx
ixixx
ixixixixx
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Example 3
![Page 19: Lesson 4-1 Polynomial Functions. Vocabulary Polynomial in one variable – an expression of the form. The coefficients represent complex numbers (real or](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649eb75503460f94bc0a29/html5/thumbnails/19.jpg)
Does the equation have an odd or even degree? How many times does the graph of
the related function cross the x-axis.
01892 23 xxx
Odd, once
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Does the equation have an odd or even degree? How many times does the graph of
the related function cross the x-axis.
03522 xxEven, twice
![Page 21: Lesson 4-1 Polynomial Functions. Vocabulary Polynomial in one variable – an expression of the form. The coefficients represent complex numbers (real or](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649eb75503460f94bc0a29/html5/thumbnails/21.jpg)
Does the equation have an odd or even degree? How many times does the graph of
the related function cross the x-axis.
02443056 2345 xxxxx
Odd, once
![Page 22: Lesson 4-1 Polynomial Functions. Vocabulary Polynomial in one variable – an expression of the form. The coefficients represent complex numbers (real or](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649eb75503460f94bc0a29/html5/thumbnails/22.jpg)
State the number of complex roots of the equation has. Then find the roots and
graph the related function.
04359 24 xx
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The polynomial has a degree of 4 so there are 4 complex roots.
2 real roots and 2 imaginary roots for a total of 4 complex roots
![Page 24: Lesson 4-1 Polynomial Functions. Vocabulary Polynomial in one variable – an expression of the form. The coefficients represent complex numbers (real or](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649eb75503460f94bc0a29/html5/thumbnails/24.jpg)
Graph with the graphing Calculator
04359 24 xx
![Page 25: Lesson 4-1 Polynomial Functions. Vocabulary Polynomial in one variable – an expression of the form. The coefficients represent complex numbers (real or](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649eb75503460f94bc0a29/html5/thumbnails/25.jpg)
State the number of complex roots of the equation has. Then find the roots and
graph the related function.
0443232 23 xxx
3 Complex Roots
Graph on graphing calculator
![Page 26: Lesson 4-1 Polynomial Functions. Vocabulary Polynomial in one variable – an expression of the form. The coefficients represent complex numbers (real or](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649eb75503460f94bc0a29/html5/thumbnails/26.jpg)
Total of 3 Complex Roots,2 imaginary, 1 real
4
2
32
4
432
)1)(432(
0)1(4)1(32
2
2
2
2
ix
x
x
xx
xxx
x-1=0x=1
0443232 23 xxx
![Page 27: Lesson 4-1 Polynomial Functions. Vocabulary Polynomial in one variable – an expression of the form. The coefficients represent complex numbers (real or](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649eb75503460f94bc0a29/html5/thumbnails/27.jpg)
State the number of complex roots of the equation has. Then find the roots and
graph the related function.
049142 xx
2 Complex Roots
Graph on graphing calculator
![Page 28: Lesson 4-1 Polynomial Functions. Vocabulary Polynomial in one variable – an expression of the form. The coefficients represent complex numbers (real or](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649eb75503460f94bc0a29/html5/thumbnails/28.jpg)
Total of 2 Complex Roots,2 real (double root)
x-7=0x=77
0)7(
)7)(7(
049142
x
x
xx
xx049142 xx
![Page 29: Lesson 4-1 Polynomial Functions. Vocabulary Polynomial in one variable – an expression of the form. The coefficients represent complex numbers (real or](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649eb75503460f94bc0a29/html5/thumbnails/29.jpg)
State the number of complex roots of the equation has. Then find the roots and
graph the related function.
082 23 xxx
3 Complex Roots
Graph on graphing calculator
![Page 30: Lesson 4-1 Polynomial Functions. Vocabulary Polynomial in one variable – an expression of the form. The coefficients represent complex numbers (real or](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649eb75503460f94bc0a29/html5/thumbnails/30.jpg)
3 complex roots3 real roots
0)2)(4(
0)82(
0822
23
xxx
xxx
xxx
x=0, x=-4, x=2
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Example 5
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