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Certainly! Here's a detailed step-by-step solution for determining the number of possible imaginary (complex) solutions for each given polynomial.
### a) [tex]\( x^5 + 13 x^4 + 36 x^3 - 148 x^2 - 832 x - 960 \)[/tex]
1. Determine the degree of the polynomial: The degree is 5.
2. A polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots (counted with multiplicity), according to the Fundamental Theorem of Algebra.
3. Identify the number of real roots.
4. The number of imaginary roots can be found by subtracting the number of real roots from the degree of the polynomial.
[tex]\[ \text{Number of real roots} = 5 \quad \text{(All roots are real)} \][/tex]
[tex]\[ \text{Number of imaginary roots} = 5 - 5 = 0 \][/tex]
### b) [tex]\( x^4 + 11 x^3 + 14 x^2 - 176 x - 480 \)[/tex]
1. Determine the degree of the polynomial: The degree is 4.
2. A polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots (counted with multiplicity).
3. Identify the number of real roots.
4. The number of imaginary roots can be found by subtracting the number of real roots from the degree of the polynomial.
[tex]\[ \text{Number of real roots} = 4 \quad \text{(All roots are real)} \][/tex]
[tex]\[ \text{Number of imaginary roots} = 4 - 4 = 0 \][/tex]
### c) [tex]\( x^2 + 10 x + 24 \)[/tex]
1. Determine the degree of the polynomial: The degree is 2.
2. A polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots (counted with multiplicity).
3. Identify the number of real roots.
4. The number of imaginary roots can be found by subtracting the number of real roots from the degree of the polynomial.
[tex]\[ \text{Number of real roots} = 2 \quad \text{(All roots are real)} \][/tex]
[tex]\[ \text{Number of imaginary roots} = 2 - 2 = 0 \][/tex]
### d) [tex]\( x^3 + 15 x^2 + 74 x + 120 \)[/tex]
1. Determine the degree of the polynomial: The degree is 3.
2. A polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots (counted with multiplicity).
3. Identify the number of real roots.
4. The number of imaginary roots can be found by subtracting the number of real roots from the degree of the polynomial.
[tex]\[ \text{Number of real roots} = 3 \quad \text{(All roots are real)} \][/tex]
[tex]\[ \text{Number of imaginary roots} = 3 - 3 = 0 \][/tex]
### e) [tex]\( -4 x - 16 \)[/tex]
1. Determine the degree of the polynomial: The degree is 1.
2. A polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots (counted with multiplicity).
3. Identify the number of real roots.
4. The number of imaginary roots can be found by subtracting the number of real roots from the degree of the polynomial.
[tex]\[ \text{Number of real roots} = 1 \quad \text{(All roots are real)} \][/tex]
[tex]\[ \text{Number of imaginary roots} = 1 - 1 = 0 \][/tex]
Therefore, the number of possible imaginary (complex) solutions for each polynomial are:
a) 0, b) 0, c) 0, d) 0, e) 0
### a) [tex]\( x^5 + 13 x^4 + 36 x^3 - 148 x^2 - 832 x - 960 \)[/tex]
1. Determine the degree of the polynomial: The degree is 5.
2. A polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots (counted with multiplicity), according to the Fundamental Theorem of Algebra.
3. Identify the number of real roots.
4. The number of imaginary roots can be found by subtracting the number of real roots from the degree of the polynomial.
[tex]\[ \text{Number of real roots} = 5 \quad \text{(All roots are real)} \][/tex]
[tex]\[ \text{Number of imaginary roots} = 5 - 5 = 0 \][/tex]
### b) [tex]\( x^4 + 11 x^3 + 14 x^2 - 176 x - 480 \)[/tex]
1. Determine the degree of the polynomial: The degree is 4.
2. A polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots (counted with multiplicity).
3. Identify the number of real roots.
4. The number of imaginary roots can be found by subtracting the number of real roots from the degree of the polynomial.
[tex]\[ \text{Number of real roots} = 4 \quad \text{(All roots are real)} \][/tex]
[tex]\[ \text{Number of imaginary roots} = 4 - 4 = 0 \][/tex]
### c) [tex]\( x^2 + 10 x + 24 \)[/tex]
1. Determine the degree of the polynomial: The degree is 2.
2. A polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots (counted with multiplicity).
3. Identify the number of real roots.
4. The number of imaginary roots can be found by subtracting the number of real roots from the degree of the polynomial.
[tex]\[ \text{Number of real roots} = 2 \quad \text{(All roots are real)} \][/tex]
[tex]\[ \text{Number of imaginary roots} = 2 - 2 = 0 \][/tex]
### d) [tex]\( x^3 + 15 x^2 + 74 x + 120 \)[/tex]
1. Determine the degree of the polynomial: The degree is 3.
2. A polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots (counted with multiplicity).
3. Identify the number of real roots.
4. The number of imaginary roots can be found by subtracting the number of real roots from the degree of the polynomial.
[tex]\[ \text{Number of real roots} = 3 \quad \text{(All roots are real)} \][/tex]
[tex]\[ \text{Number of imaginary roots} = 3 - 3 = 0 \][/tex]
### e) [tex]\( -4 x - 16 \)[/tex]
1. Determine the degree of the polynomial: The degree is 1.
2. A polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots (counted with multiplicity).
3. Identify the number of real roots.
4. The number of imaginary roots can be found by subtracting the number of real roots from the degree of the polynomial.
[tex]\[ \text{Number of real roots} = 1 \quad \text{(All roots are real)} \][/tex]
[tex]\[ \text{Number of imaginary roots} = 1 - 1 = 0 \][/tex]
Therefore, the number of possible imaginary (complex) solutions for each polynomial are:
a) 0, b) 0, c) 0, d) 0, e) 0
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