Answer :
To write a polynomial function of the least degree with real coefficients that has the given zeros [tex]\( -2, -4, -3 + 4i \)[/tex], and [tex]\( -3 - 4i \)[/tex], we must utilize the fact that complex zeros always come in conjugate pairs when the polynomial has real coefficients. Let's break down the steps to form the polynomial and derive the standard form:
1. List the given zeros:
[tex]\[ -2, -4, -3 + 4i, -3 - 4i \][/tex]
2. Form factors from each zero:
For each given zero [tex]\( r \)[/tex], a corresponding factor is [tex]\( (x - r) \)[/tex].
[tex]\[ (x + 2) \quad \text{for} \quad -2 \][/tex]
[tex]\[ (x + 4) \quad \text{for} \quad -4 \][/tex]
[tex]\[ (x - (-3 + 4i)) = (x + 3 - 4i) \quad \text{for} \quad -3+4i \][/tex]
[tex]\[ (x - (-3 - 4i)) = (x + 3 + 4i) \quad \text{for} \quad -3-4i \][/tex]
3. Multiply the conjugate pairs first:
Multiply the factors corresponding to the complex zeros [tex]\( -3 + 4i \)[/tex] and [tex]\( -3 - 4i \)[/tex].
[tex]\[ (x + 3 - 4i)(x + 3 + 4i) \][/tex]
This can be simplified using the difference of squares:
[tex]\[ (x + 3)^2 - (4i)^2 \][/tex]
[tex]\[ (x + 3)^2 - 16i^2 \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ (x + 3)^2 + 16 \][/tex]
[tex]\[ x^2 + 6x + 9 + 16 \][/tex]
[tex]\[ x^2 + 6x + 25 \][/tex]
4. Combine all the factored terms:
The remaining factors are [tex]\( (x + 2) \)[/tex] and [tex]\( (x + 4) \)[/tex].
[tex]\[ (x + 2)(x + 4) \][/tex]
5. Construct the full polynomial:
Multiply out the factors we have to get the polynomial of least degree:
[tex]\[ (x + 2)(x + 4)(x^2 + 6x + 25) \][/tex]
6. Expand the polynomial:
First multiply [tex]\( (x + 2) \)[/tex] and [tex]\( (x + 4) \)[/tex]:
[tex]\[ (x + 2)(x + 4) = x^2 + 6x + 8 \][/tex]
Then multiply [tex]\( (x^2 + 6x + 8) \)[/tex] with [tex]\( (x^2 + 6x + 25) \)[/tex]:
To ensure clarity, multiply each term systematically:
[tex]\[ (x^2 + 6x + 8)(x^2 + 6x + 25) \][/tex]
Distribute [tex]\( x^2 \)[/tex] to each term in [tex]\( (x^2 + 6x + 25) \)[/tex]:
[tex]\[ x^4 + 6x^3 + 25x^2 \][/tex]
Distribute [tex]\( 6x \)[/tex] to each term in [tex]\( (x^2 + 6x + 25) \)[/tex]:
[tex]\[ + 6x^3 + 36x^2 + 150x \][/tex]
Distribute [tex]\( 8 \)[/tex] to each term in [tex]\( (x^2 + 6x + 25) \)[/tex]:
[tex]\[ + 8x^2 + 48x + 200 \][/tex]
Combine like terms:
[tex]\[ x^4 + 12x^3 + (25x^2 + 36x^2 + 8x^2) + 198x + 200 \][/tex]
[tex]\[ x^4 + 12x^3 + 69x^2 + 198x + 200 \][/tex]
Therefore, the polynomial of least degree with real coefficients that has the given zeros [tex]\( -2, -4, -3 + 4i, \)[/tex] and [tex]\( -3 - 4i \)[/tex] in standard form is:
[tex]\[ \boxed{x^4 + 12x^3 + 69x^2 + 198x + 200} \][/tex]
1. List the given zeros:
[tex]\[ -2, -4, -3 + 4i, -3 - 4i \][/tex]
2. Form factors from each zero:
For each given zero [tex]\( r \)[/tex], a corresponding factor is [tex]\( (x - r) \)[/tex].
[tex]\[ (x + 2) \quad \text{for} \quad -2 \][/tex]
[tex]\[ (x + 4) \quad \text{for} \quad -4 \][/tex]
[tex]\[ (x - (-3 + 4i)) = (x + 3 - 4i) \quad \text{for} \quad -3+4i \][/tex]
[tex]\[ (x - (-3 - 4i)) = (x + 3 + 4i) \quad \text{for} \quad -3-4i \][/tex]
3. Multiply the conjugate pairs first:
Multiply the factors corresponding to the complex zeros [tex]\( -3 + 4i \)[/tex] and [tex]\( -3 - 4i \)[/tex].
[tex]\[ (x + 3 - 4i)(x + 3 + 4i) \][/tex]
This can be simplified using the difference of squares:
[tex]\[ (x + 3)^2 - (4i)^2 \][/tex]
[tex]\[ (x + 3)^2 - 16i^2 \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ (x + 3)^2 + 16 \][/tex]
[tex]\[ x^2 + 6x + 9 + 16 \][/tex]
[tex]\[ x^2 + 6x + 25 \][/tex]
4. Combine all the factored terms:
The remaining factors are [tex]\( (x + 2) \)[/tex] and [tex]\( (x + 4) \)[/tex].
[tex]\[ (x + 2)(x + 4) \][/tex]
5. Construct the full polynomial:
Multiply out the factors we have to get the polynomial of least degree:
[tex]\[ (x + 2)(x + 4)(x^2 + 6x + 25) \][/tex]
6. Expand the polynomial:
First multiply [tex]\( (x + 2) \)[/tex] and [tex]\( (x + 4) \)[/tex]:
[tex]\[ (x + 2)(x + 4) = x^2 + 6x + 8 \][/tex]
Then multiply [tex]\( (x^2 + 6x + 8) \)[/tex] with [tex]\( (x^2 + 6x + 25) \)[/tex]:
To ensure clarity, multiply each term systematically:
[tex]\[ (x^2 + 6x + 8)(x^2 + 6x + 25) \][/tex]
Distribute [tex]\( x^2 \)[/tex] to each term in [tex]\( (x^2 + 6x + 25) \)[/tex]:
[tex]\[ x^4 + 6x^3 + 25x^2 \][/tex]
Distribute [tex]\( 6x \)[/tex] to each term in [tex]\( (x^2 + 6x + 25) \)[/tex]:
[tex]\[ + 6x^3 + 36x^2 + 150x \][/tex]
Distribute [tex]\( 8 \)[/tex] to each term in [tex]\( (x^2 + 6x + 25) \)[/tex]:
[tex]\[ + 8x^2 + 48x + 200 \][/tex]
Combine like terms:
[tex]\[ x^4 + 12x^3 + (25x^2 + 36x^2 + 8x^2) + 198x + 200 \][/tex]
[tex]\[ x^4 + 12x^3 + 69x^2 + 198x + 200 \][/tex]
Therefore, the polynomial of least degree with real coefficients that has the given zeros [tex]\( -2, -4, -3 + 4i, \)[/tex] and [tex]\( -3 - 4i \)[/tex] in standard form is:
[tex]\[ \boxed{x^4 + 12x^3 + 69x^2 + 198x + 200} \][/tex]
