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Answer :
To find the zeros of the polynomial function [tex]\( f(x) = x^4 + 3x^3 - 21x^2 - 81x - 70 \)[/tex], we can use algebraic techniques to factor and solve for the values of [tex]\( x \)[/tex] that make the function equal to zero. Here is a step-by-step guide:
1. Check for Rational Roots:
Start by checking for any obvious rational roots using the Rational Root Theorem. This theorem suggests that any rational root, in the form [tex]\( \frac{p}{q} \)[/tex], is a factor of the constant term (-70) divided by a factor of the leading coefficient (1). Possible rational roots include ±1, ±2, ±5, ±7, ±10, ±14, ±35, and ±70.
2. Synthetic Division or Factor Testing:
Test these possible roots by substituting them into the polynomial to see if they result in zero. For example:
- Test [tex]\( x = -2 \)[/tex]: Substituting back into the polynomial results in zero, meaning [tex]\( x + 2 \)[/tex] is a factor.
- Test [tex]\( x = 5 \)[/tex]: Substituting back also results in zero, meaning [tex]\( x - 5 \)[/tex] is a factor.
3. Polynomial Division:
After finding the roots [tex]\( x = -2 \)[/tex] and [tex]\( x = 5 \)[/tex], divide the polynomial by [tex]\( (x + 2)(x - 5) \)[/tex] to simplify it further.
4. Solve the Remaining Quadratic:
The division results in a quadratic equation. Solve the quadratic equation using the quadratic formula, factoring, or completing the square if applicable.
5. Finding Remaining Roots:
This process will uncover additional roots. In our solution, these are [tex]\( x = -3 - \sqrt{2} \)[/tex] and [tex]\( x = -3 + \sqrt{2} \)[/tex].
Therefore, the zeros of the polynomial [tex]\( f(x) \)[/tex] are [tex]\( -2, 5, -3 - \sqrt{2}, -3 + \sqrt{2} \)[/tex]. These are the exact values of the roots without using any decimal approximations.
1. Check for Rational Roots:
Start by checking for any obvious rational roots using the Rational Root Theorem. This theorem suggests that any rational root, in the form [tex]\( \frac{p}{q} \)[/tex], is a factor of the constant term (-70) divided by a factor of the leading coefficient (1). Possible rational roots include ±1, ±2, ±5, ±7, ±10, ±14, ±35, and ±70.
2. Synthetic Division or Factor Testing:
Test these possible roots by substituting them into the polynomial to see if they result in zero. For example:
- Test [tex]\( x = -2 \)[/tex]: Substituting back into the polynomial results in zero, meaning [tex]\( x + 2 \)[/tex] is a factor.
- Test [tex]\( x = 5 \)[/tex]: Substituting back also results in zero, meaning [tex]\( x - 5 \)[/tex] is a factor.
3. Polynomial Division:
After finding the roots [tex]\( x = -2 \)[/tex] and [tex]\( x = 5 \)[/tex], divide the polynomial by [tex]\( (x + 2)(x - 5) \)[/tex] to simplify it further.
4. Solve the Remaining Quadratic:
The division results in a quadratic equation. Solve the quadratic equation using the quadratic formula, factoring, or completing the square if applicable.
5. Finding Remaining Roots:
This process will uncover additional roots. In our solution, these are [tex]\( x = -3 - \sqrt{2} \)[/tex] and [tex]\( x = -3 + \sqrt{2} \)[/tex].
Therefore, the zeros of the polynomial [tex]\( f(x) \)[/tex] are [tex]\( -2, 5, -3 - \sqrt{2}, -3 + \sqrt{2} \)[/tex]. These are the exact values of the roots without using any decimal approximations.
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