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Answer :
To find a polynomial function of least degree with integral coefficients that has the zeros [tex]\(2, -2, 3, -4\)[/tex], we can follow these steps:
1. Start with the Zeros:
The given zeros are [tex]\(2\)[/tex], [tex]\(-2\)[/tex], [tex]\(3\)[/tex], and [tex]\(-4\)[/tex].
2. Form the Factors:
For each zero, we can form a factor of the polynomial by setting each zero in the form [tex]\((x - \text{zero})\)[/tex]. Thus, the factors from the zeros are:
- For [tex]\(2\)[/tex], the factor is [tex]\((x - 2)\)[/tex].
- For [tex]\(-2\)[/tex], the factor is [tex]\((x + 2)\)[/tex].
- For [tex]\(3\)[/tex], the factor is [tex]\((x - 3)\)[/tex].
- For [tex]\(-4\)[/tex], the factor is [tex]\((x + 4)\)[/tex].
3. Construct the Polynomial:
Multiply all the factors together to form the polynomial:
[tex]\[
(x - 2)(x + 2)(x - 3)(x + 4)
\][/tex]
4. Multiply the Pairs:
- First, multiply [tex]\((x - 2)\)[/tex] and [tex]\((x + 2)\)[/tex]:
[tex]\[
(x - 2)(x + 2) = x^2 - 4
\][/tex]
- Then, multiply [tex]\((x - 3)\)[/tex] and [tex]\((x + 4)\)[/tex]:
[tex]\[
(x - 3)(x + 4) = x^2 + x - 12
\][/tex]
5. Combine the Results:
Now, multiply the two quadratic expressions obtained:
[tex]\[
(x^2 - 4)(x^2 + x - 12)
\][/tex]
6. Expand the Product:
Distribute the multiplication:
[tex]\[
\begin{align*}
x^2(x^2 + x - 12) & = x^4 + x^3 - 12x^2, \\
-4(x^2 + x - 12) & = -4x^2 - 4x + 48.
\end{align*}
\][/tex]
Combine the terms:
[tex]\[
x^4 + x^3 - 12x^2 - 4x^2 - 4x + 48
\][/tex]
7. Simplify the Polynomial:
Now, add the like terms:
[tex]\[
x^4 + x^3 - 16x^2 - 4x + 48
\][/tex]
So, the polynomial of least degree with the given zeros and integral coefficients is:
[tex]\[ x^4 + x^3 - 16x^2 - 4x + 48 \][/tex]
1. Start with the Zeros:
The given zeros are [tex]\(2\)[/tex], [tex]\(-2\)[/tex], [tex]\(3\)[/tex], and [tex]\(-4\)[/tex].
2. Form the Factors:
For each zero, we can form a factor of the polynomial by setting each zero in the form [tex]\((x - \text{zero})\)[/tex]. Thus, the factors from the zeros are:
- For [tex]\(2\)[/tex], the factor is [tex]\((x - 2)\)[/tex].
- For [tex]\(-2\)[/tex], the factor is [tex]\((x + 2)\)[/tex].
- For [tex]\(3\)[/tex], the factor is [tex]\((x - 3)\)[/tex].
- For [tex]\(-4\)[/tex], the factor is [tex]\((x + 4)\)[/tex].
3. Construct the Polynomial:
Multiply all the factors together to form the polynomial:
[tex]\[
(x - 2)(x + 2)(x - 3)(x + 4)
\][/tex]
4. Multiply the Pairs:
- First, multiply [tex]\((x - 2)\)[/tex] and [tex]\((x + 2)\)[/tex]:
[tex]\[
(x - 2)(x + 2) = x^2 - 4
\][/tex]
- Then, multiply [tex]\((x - 3)\)[/tex] and [tex]\((x + 4)\)[/tex]:
[tex]\[
(x - 3)(x + 4) = x^2 + x - 12
\][/tex]
5. Combine the Results:
Now, multiply the two quadratic expressions obtained:
[tex]\[
(x^2 - 4)(x^2 + x - 12)
\][/tex]
6. Expand the Product:
Distribute the multiplication:
[tex]\[
\begin{align*}
x^2(x^2 + x - 12) & = x^4 + x^3 - 12x^2, \\
-4(x^2 + x - 12) & = -4x^2 - 4x + 48.
\end{align*}
\][/tex]
Combine the terms:
[tex]\[
x^4 + x^3 - 12x^2 - 4x^2 - 4x + 48
\][/tex]
7. Simplify the Polynomial:
Now, add the like terms:
[tex]\[
x^4 + x^3 - 16x^2 - 4x + 48
\][/tex]
So, the polynomial of least degree with the given zeros and integral coefficients is:
[tex]\[ x^4 + x^3 - 16x^2 - 4x + 48 \][/tex]
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Rewritten by : Barada
