Answer :
To find the zeros of the polynomial function [tex]\( f(x) = x^4 - 4x^3 - 14x^2 + 36x + 45 \)[/tex], we'll determine what values of [tex]\( x \)[/tex] make the function equal to zero. In simpler terms, we need to solve the equation [tex]\( f(x) = 0 \)[/tex].
Here is a step-by-step approach:
1. Understanding the Problem: We are working with a polynomial of degree 4, meaning there are at most 4 real roots.
2. Look for Rational Roots by Testing Small Integers: Start testing some small integer values that might be roots (meaning they simplify the polynomial to zero). This could involve plugging values into the polynomial.
3. Factor the Polynomial: Once we determine that certain values are roots, we factor the polynomial accordingly. When a root, [tex]\( r \)[/tex], is found, [tex]\( (x - r) \)[/tex] becomes a factor of the polynomial.
4. Use Algebraic Techniques or Tools for Solving Polynomials: When dealing with higher-degree polynomials, sometimes it's helpful to use algebraic tools or calculators to find the roots. These approaches can provide exact or approximate roots if the factoring directly is complex.
5. Verify the Roots: Once potential roots are found, plug these values back into the original polynomial to ensure that they indeed result in zero.
Upon following these steps, we find that the zeros of the polynomial [tex]\( f(x) = x^4 - 4x^3 - 14x^2 + 36x + 45 \)[/tex] are:
- [tex]\( x = -3 \)[/tex]
- [tex]\( x = -1 \)[/tex]
- [tex]\( x = 3 \)[/tex]
- [tex]\( x = 5 \)[/tex]
These are the values that make the function equal to zero, meaning they are the points where the graph of the polynomial crosses or touches the x-axis.
Here is a step-by-step approach:
1. Understanding the Problem: We are working with a polynomial of degree 4, meaning there are at most 4 real roots.
2. Look for Rational Roots by Testing Small Integers: Start testing some small integer values that might be roots (meaning they simplify the polynomial to zero). This could involve plugging values into the polynomial.
3. Factor the Polynomial: Once we determine that certain values are roots, we factor the polynomial accordingly. When a root, [tex]\( r \)[/tex], is found, [tex]\( (x - r) \)[/tex] becomes a factor of the polynomial.
4. Use Algebraic Techniques or Tools for Solving Polynomials: When dealing with higher-degree polynomials, sometimes it's helpful to use algebraic tools or calculators to find the roots. These approaches can provide exact or approximate roots if the factoring directly is complex.
5. Verify the Roots: Once potential roots are found, plug these values back into the original polynomial to ensure that they indeed result in zero.
Upon following these steps, we find that the zeros of the polynomial [tex]\( f(x) = x^4 - 4x^3 - 14x^2 + 36x + 45 \)[/tex] are:
- [tex]\( x = -3 \)[/tex]
- [tex]\( x = -1 \)[/tex]
- [tex]\( x = 3 \)[/tex]
- [tex]\( x = 5 \)[/tex]
These are the values that make the function equal to zero, meaning they are the points where the graph of the polynomial crosses or touches the x-axis.
