Answer :
To write a polynomial function of least degree with real coefficients in standard form that has the given zeros: [tex]\(-2\)[/tex], [tex]\(-4\)[/tex], and [tex]\(-3+4i\)[/tex], follow these step-by-step instructions:
1. Identify the zeros and their conjugates:
- Zeros: [tex]\(-2\)[/tex], [tex]\(-4\)[/tex], and [tex]\(-3+4i\)[/tex].
- Since the coefficients must be real, the conjugate of the complex zero [tex]\(-3+4i\)[/tex] must also be included. Thus, the conjugate is [tex]\(-3-4i\)[/tex].
2. Express each zero as a factor:
- For each zero [tex]\(a\)[/tex], the corresponding factor is [tex]\((x - a)\)[/tex].
- Therefore, the factors are:
[tex]\[ (x + 2), (x + 4), (x + 3 - 4i), \text{ and } (x + 3 + 4i) \][/tex]
3. Combine the complex conjugate factors into a quadratic factor:
- Multiply the conjugate pair:
[tex]\[ (x + 3 - 4i)(x + 3 + 4i) \][/tex]
- This can be simplified using the difference of squares:
[tex]\[ (x + 3)^2 - (4i)^2 \][/tex]
- Calculating inside:
[tex]\[ (x + 3)^2 = x^2 + 6x + 9 \][/tex]
[tex]\[ (4i)^2 = 16i^2 = -16 \][/tex]
- Putting it together:
[tex]\[ (x + 3)^2 - (4i)^2 = x^2 + 6x + 9 - (-16) = x^2 + 6x + 25 \][/tex]
4. Multiply all the factors together:
- Now multiply the remaining linear factors by this quadratic factor:
[tex]\[ (x + 2)(x + 4)(x^2 + 6x + 25) \][/tex]
5. First, multiply the linear factors:
- Compute [tex]\((x + 2)(x + 4)\)[/tex]:
[tex]\[ (x + 2)(x + 4) = x^2 + 4x + 2x + 8 = x^2 + 6x + 8 \][/tex]
6. Next, multiply this result by the quadratic factor:
- Compute [tex]\((x^2 + 6x + 8)(x^2 + 6x + 25)\)[/tex]:
[tex]\[ (x^2 + 6x + 8)(x^2 + 6x + 25) \][/tex]
7. Perform the polynomial multiplication:
- Multiply each term in [tex]\((x^2 + 6x + 8)\)[/tex] by each term in [tex]\((x^2 + 6x + 25)\)[/tex]:
- [tex]\(x^2 \cdot x^2 = x^4\)[/tex]
- [tex]\(x^2 \cdot 6x = 6x^3\)[/tex]
- [tex]\(x^2 \cdot 25 = 25x^2\)[/tex]
- [tex]\(6x \cdot x^2 = 6x^3\)[/tex]
- [tex]\(6x \cdot 6x = 36x^2\)[/tex]
- [tex]\(6x \cdot 25 = 150x\)[/tex]
- [tex]\(8 \cdot x^2 = 8x^2\)[/tex]
- [tex]\(8 \cdot 6x = 48x\)[/tex]
- [tex]\(8 \cdot 25 = 200\)[/tex]
- Summing all these terms:
[tex]\[ x^4 + 12x^3 + (25x^2 + 36x^2 + 8x^2) + (150x + 48x) + 200 \][/tex]
- Combine like terms:
[tex]\[ x^4 + 12x^3 + 69x^2 + 198x + 200 \][/tex]
Therefore, the polynomial function of least degree with real coefficients in standard form that has the given zeros is:
[tex]\[ \boxed{x^4 + 12x^3 + 69x^2 + 198x + 200} \][/tex]
1. Identify the zeros and their conjugates:
- Zeros: [tex]\(-2\)[/tex], [tex]\(-4\)[/tex], and [tex]\(-3+4i\)[/tex].
- Since the coefficients must be real, the conjugate of the complex zero [tex]\(-3+4i\)[/tex] must also be included. Thus, the conjugate is [tex]\(-3-4i\)[/tex].
2. Express each zero as a factor:
- For each zero [tex]\(a\)[/tex], the corresponding factor is [tex]\((x - a)\)[/tex].
- Therefore, the factors are:
[tex]\[ (x + 2), (x + 4), (x + 3 - 4i), \text{ and } (x + 3 + 4i) \][/tex]
3. Combine the complex conjugate factors into a quadratic factor:
- Multiply the conjugate pair:
[tex]\[ (x + 3 - 4i)(x + 3 + 4i) \][/tex]
- This can be simplified using the difference of squares:
[tex]\[ (x + 3)^2 - (4i)^2 \][/tex]
- Calculating inside:
[tex]\[ (x + 3)^2 = x^2 + 6x + 9 \][/tex]
[tex]\[ (4i)^2 = 16i^2 = -16 \][/tex]
- Putting it together:
[tex]\[ (x + 3)^2 - (4i)^2 = x^2 + 6x + 9 - (-16) = x^2 + 6x + 25 \][/tex]
4. Multiply all the factors together:
- Now multiply the remaining linear factors by this quadratic factor:
[tex]\[ (x + 2)(x + 4)(x^2 + 6x + 25) \][/tex]
5. First, multiply the linear factors:
- Compute [tex]\((x + 2)(x + 4)\)[/tex]:
[tex]\[ (x + 2)(x + 4) = x^2 + 4x + 2x + 8 = x^2 + 6x + 8 \][/tex]
6. Next, multiply this result by the quadratic factor:
- Compute [tex]\((x^2 + 6x + 8)(x^2 + 6x + 25)\)[/tex]:
[tex]\[ (x^2 + 6x + 8)(x^2 + 6x + 25) \][/tex]
7. Perform the polynomial multiplication:
- Multiply each term in [tex]\((x^2 + 6x + 8)\)[/tex] by each term in [tex]\((x^2 + 6x + 25)\)[/tex]:
- [tex]\(x^2 \cdot x^2 = x^4\)[/tex]
- [tex]\(x^2 \cdot 6x = 6x^3\)[/tex]
- [tex]\(x^2 \cdot 25 = 25x^2\)[/tex]
- [tex]\(6x \cdot x^2 = 6x^3\)[/tex]
- [tex]\(6x \cdot 6x = 36x^2\)[/tex]
- [tex]\(6x \cdot 25 = 150x\)[/tex]
- [tex]\(8 \cdot x^2 = 8x^2\)[/tex]
- [tex]\(8 \cdot 6x = 48x\)[/tex]
- [tex]\(8 \cdot 25 = 200\)[/tex]
- Summing all these terms:
[tex]\[ x^4 + 12x^3 + (25x^2 + 36x^2 + 8x^2) + (150x + 48x) + 200 \][/tex]
- Combine like terms:
[tex]\[ x^4 + 12x^3 + 69x^2 + 198x + 200 \][/tex]
Therefore, the polynomial function of least degree with real coefficients in standard form that has the given zeros is:
[tex]\[ \boxed{x^4 + 12x^3 + 69x^2 + 198x + 200} \][/tex]
