Answer :
To write a polynomial of least degree with integral coefficients that has the given zeros [tex]\(-5, 4, 4-i, 4+i\)[/tex], let's follow these steps:
Step 1: Understand the Zeros
- Zeros given are [tex]\(-5, 4, 4-i, 4+i\)[/tex].
- Notice that complex roots appear in conjugate pairs. This is important for forming a polynomial with real coefficients.
Step 2: Form Factors from Zeros
- Each zero contributes a factor of the form [tex]\(x - \text{zero}\)[/tex].
- So our factors are:
- [tex]\(x - (-5) = x + 5\)[/tex]
- [tex]\(x - 4\)[/tex]
- [tex]\(x - (4 - i)\)[/tex]
- [tex]\(x - (4 + i)\)[/tex]
Step 3: Simplify Complex Factors
- Pair the complex conjugates:
[tex]\[
(x - (4-i))(x - (4+i))
\][/tex]
- Use the formula [tex]\((a-b)(a+b) = a^2 - b^2\)[/tex]:
[tex]\[
(x - 4 + i)(x - 4 - i) = ((x - 4) + i)((x - 4) - i) = (x - 4)^2 - i^2
\][/tex]
- Recall [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
(x - 4)^2 - (-1) = (x - 4)^2 + 1
\][/tex]
- Expand [tex]\((x - 4)^2\)[/tex]:
[tex]\[
(x - 4)^2 = x^2 - 8x + 16
\][/tex]
- Therefore, [tex]\((x - (4-i))(x - (4+i)) = x^2 - 8x + 17\)[/tex].
Step 4: Construct the Polynomial
- Combine all factors to form the polynomial:
[tex]\[
(x + 5)(x - 4)(x^2 - 8x + 17)
\][/tex]
Step 5: Expand Polynomial
1. First, expand [tex]\((x + 5)(x - 4)\)[/tex]:
[tex]\[
(x + 5)(x - 4) = x^2 - 4x + 5x - 20 = x^2 + x - 20
\][/tex]
2. Multiply this result by [tex]\((x^2 - 8x + 17)\)[/tex]:
- Calculate [tex]\((x^2 + x - 20)(x^2 - 8x + 17)\)[/tex].
Expanding step by step:
- [tex]\((x^2)(x^2) = x^4\)[/tex]
- [tex]\((x^2)(-8x) = -8x^3\)[/tex]
- [tex]\((x^2)(17) = 17x^2\)[/tex]
- [tex]\((x)(x^2) = x^3\)[/tex]
- [tex]\((x)(-8x) = -8x^2\)[/tex]
- [tex]\((x)(17) = 17x\)[/tex]
- [tex]\((-20)(x^2) = -20x^2\)[/tex]
- [tex]\((-20)(-8x) = 160x\)[/tex]
- [tex]\((-20)(17) = -340\)[/tex]
Now combine these:
[tex]\[
x^4 - 8x^3 + 17x^2 + x^3 - 8x^2 + 17x - 20x^2 + 160x - 340
\][/tex]
Combine like terms:
- The [tex]\(x^4\)[/tex] term is [tex]\(x^4\)[/tex].
- The [tex]\(x^3\)[/tex] terms: [tex]\(-8x^3 + x^3 = -7x^3\)[/tex].
- The [tex]\(x^2\)[/tex] terms: [tex]\(17x^2 - 8x^2 - 20x^2 = -11x^2\)[/tex].
- The [tex]\(x\)[/tex] terms: [tex]\(17x + 160x = 177x\)[/tex].
- The constant term is [tex]\(-340\)[/tex].
Thus, the polynomial of least degree with integral coefficients is:
[tex]\[
x^4 - 7x^3 - 11x^2 + 177x - 340
\][/tex]
Step 1: Understand the Zeros
- Zeros given are [tex]\(-5, 4, 4-i, 4+i\)[/tex].
- Notice that complex roots appear in conjugate pairs. This is important for forming a polynomial with real coefficients.
Step 2: Form Factors from Zeros
- Each zero contributes a factor of the form [tex]\(x - \text{zero}\)[/tex].
- So our factors are:
- [tex]\(x - (-5) = x + 5\)[/tex]
- [tex]\(x - 4\)[/tex]
- [tex]\(x - (4 - i)\)[/tex]
- [tex]\(x - (4 + i)\)[/tex]
Step 3: Simplify Complex Factors
- Pair the complex conjugates:
[tex]\[
(x - (4-i))(x - (4+i))
\][/tex]
- Use the formula [tex]\((a-b)(a+b) = a^2 - b^2\)[/tex]:
[tex]\[
(x - 4 + i)(x - 4 - i) = ((x - 4) + i)((x - 4) - i) = (x - 4)^2 - i^2
\][/tex]
- Recall [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
(x - 4)^2 - (-1) = (x - 4)^2 + 1
\][/tex]
- Expand [tex]\((x - 4)^2\)[/tex]:
[tex]\[
(x - 4)^2 = x^2 - 8x + 16
\][/tex]
- Therefore, [tex]\((x - (4-i))(x - (4+i)) = x^2 - 8x + 17\)[/tex].
Step 4: Construct the Polynomial
- Combine all factors to form the polynomial:
[tex]\[
(x + 5)(x - 4)(x^2 - 8x + 17)
\][/tex]
Step 5: Expand Polynomial
1. First, expand [tex]\((x + 5)(x - 4)\)[/tex]:
[tex]\[
(x + 5)(x - 4) = x^2 - 4x + 5x - 20 = x^2 + x - 20
\][/tex]
2. Multiply this result by [tex]\((x^2 - 8x + 17)\)[/tex]:
- Calculate [tex]\((x^2 + x - 20)(x^2 - 8x + 17)\)[/tex].
Expanding step by step:
- [tex]\((x^2)(x^2) = x^4\)[/tex]
- [tex]\((x^2)(-8x) = -8x^3\)[/tex]
- [tex]\((x^2)(17) = 17x^2\)[/tex]
- [tex]\((x)(x^2) = x^3\)[/tex]
- [tex]\((x)(-8x) = -8x^2\)[/tex]
- [tex]\((x)(17) = 17x\)[/tex]
- [tex]\((-20)(x^2) = -20x^2\)[/tex]
- [tex]\((-20)(-8x) = 160x\)[/tex]
- [tex]\((-20)(17) = -340\)[/tex]
Now combine these:
[tex]\[
x^4 - 8x^3 + 17x^2 + x^3 - 8x^2 + 17x - 20x^2 + 160x - 340
\][/tex]
Combine like terms:
- The [tex]\(x^4\)[/tex] term is [tex]\(x^4\)[/tex].
- The [tex]\(x^3\)[/tex] terms: [tex]\(-8x^3 + x^3 = -7x^3\)[/tex].
- The [tex]\(x^2\)[/tex] terms: [tex]\(17x^2 - 8x^2 - 20x^2 = -11x^2\)[/tex].
- The [tex]\(x\)[/tex] terms: [tex]\(17x + 160x = 177x\)[/tex].
- The constant term is [tex]\(-340\)[/tex].
Thus, the polynomial of least degree with integral coefficients is:
[tex]\[
x^4 - 7x^3 - 11x^2 + 177x - 340
\][/tex]
