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Answer :
Sure! Let's solve the polynomial equation [tex]\( f(x) = x^4 - 9x^3 + 15x^2 + 39x - 70 \)[/tex] to find its zeros. Here’s the detailed step-by-step solution:
1. Equation:
[tex]\[
f(x) = x^4 - 9x^3 + 15x^2 + 39x - 70
\][/tex]
2. Identify Potential Rational Zeros:
Use the Rational Root Theorem, which states that any rational solution, p/q, of the polynomial equation is a factor of the constant term divided by a factor of the leading coefficient.
For this polynomial:
- Constant term: [tex]\(-70\)[/tex]
- Leading coefficient: [tex]\(1\)[/tex]
Potential rational roots could be [tex]\(\pm 1, \pm 2, \pm 5, \pm 7, \pm 10, \pm 14, \pm 35, \pm 70\)[/tex].
3. Testing Potential Rational Zeros:
- After testing, we find that [tex]\( x = -2 \)[/tex] and [tex]\( x = 5 \)[/tex] are zeros of the polynomial.
4. Factor the Polynomial:
We can factor the polynomial by using these zeros:
[tex]\[
f(x) = (x + 2)(x - 5)Q(x)
\][/tex]
where [tex]\(Q(x)\)[/tex] is a quadratic polynomial obtained by synthetic or polynomial division.
5. Find the Remaining Zeros:
We then solve the quadratic factor [tex]\(Q(x)\)[/tex] to find the remaining zeros. This factorization leads to:
[tex]\[
(x + 2)(x - 5)(x^2 - 6x + 7) = 0
\][/tex]
Solving the quadratic equation [tex]\( x^2 - 6x + 7 = 0 \)[/tex] gives the remaining zeros:
[tex]\[
x = 3 \pm \sqrt{2}
\][/tex]
6. List All Zeros:
Combining all the results, the zeros of the polynomial are:
[tex]\[
-2, 5, 3 - \sqrt{2}, 3 + \sqrt{2}
\][/tex]
So, the exact values of these zeros are [tex]\(-2, 5, 3 - \sqrt{2}, 3 + \sqrt{2}\)[/tex].
1. Equation:
[tex]\[
f(x) = x^4 - 9x^3 + 15x^2 + 39x - 70
\][/tex]
2. Identify Potential Rational Zeros:
Use the Rational Root Theorem, which states that any rational solution, p/q, of the polynomial equation is a factor of the constant term divided by a factor of the leading coefficient.
For this polynomial:
- Constant term: [tex]\(-70\)[/tex]
- Leading coefficient: [tex]\(1\)[/tex]
Potential rational roots could be [tex]\(\pm 1, \pm 2, \pm 5, \pm 7, \pm 10, \pm 14, \pm 35, \pm 70\)[/tex].
3. Testing Potential Rational Zeros:
- After testing, we find that [tex]\( x = -2 \)[/tex] and [tex]\( x = 5 \)[/tex] are zeros of the polynomial.
4. Factor the Polynomial:
We can factor the polynomial by using these zeros:
[tex]\[
f(x) = (x + 2)(x - 5)Q(x)
\][/tex]
where [tex]\(Q(x)\)[/tex] is a quadratic polynomial obtained by synthetic or polynomial division.
5. Find the Remaining Zeros:
We then solve the quadratic factor [tex]\(Q(x)\)[/tex] to find the remaining zeros. This factorization leads to:
[tex]\[
(x + 2)(x - 5)(x^2 - 6x + 7) = 0
\][/tex]
Solving the quadratic equation [tex]\( x^2 - 6x + 7 = 0 \)[/tex] gives the remaining zeros:
[tex]\[
x = 3 \pm \sqrt{2}
\][/tex]
6. List All Zeros:
Combining all the results, the zeros of the polynomial are:
[tex]\[
-2, 5, 3 - \sqrt{2}, 3 + \sqrt{2}
\][/tex]
So, the exact values of these zeros are [tex]\(-2, 5, 3 - \sqrt{2}, 3 + \sqrt{2}\)[/tex].
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