To find the zeros of the polynomial function f(x) = 5x^3 – 2x^2 + 8x – 3, we must first set the function equal to zero:
5x^3 – 2x^2 + 8x – 3 = 0
This is a cubic polynomial, which can potentially have up to three real zeros. We can use various methods to find the zeros, such as the Rational Root Theorem, synthetic division, or graphing calculators. For this example, let’s start with the Rational Root Theorem.
The Rational Root Theorem suggests that any rational zero, in the form of p/q (where p is a factor of the constant term and q is a factor of the leading coefficient), can be used to test possible solutions.
The factors of the constant term (-3) are ±1, ±3, and the factors of the leading coefficient (5) are ±1, ±5. Thus, the possible rational zeros could be:
- ±1, ±3, ±1/5, ±3/5
Next, we can test these possible zeros by substituting them into the polynomial:
- When we test x = 1:
f(1) = 5(1)^3 – 2(1)^2 + 8(1) – 3 = 5 – 2 + 8 – 3 = 8 (not a zero)
- When we test x = -1:
f(-1) = 5(-1)^3 – 2(-1)^2 + 8(-1) – 3 = -5 – 2 – 8 – 3 = -18 (not a zero)
- When we test x = 3:
f(3) = 5(3)^3 – 2(3)^2 + 8(3) – 3 = 5(27) – 2(9) + 24 – 3 = 135 – 18 + 24 – 3 = 138 (not a zero)
- When we test x = -3:
f(-3) = 5(-3)^3 – 2(-3)^2 + 8(-3) – 3 = 5(-27) – 2(9) – 24 – 3 = -135 – 18 – 24 – 3 = -180 (not a zero)
- When we test x = 1/5:
f(1/5) = 5(1/5)^3 – 2(1/5)^2 + 8(1/5) – 3 = 5(1/125) – 2(1/25) + 8(1/5) – 3 = 0 (this may be a zero)
Now that we have found one real zero, we can perform synthetic division or polynomial long division to reduce the degree of the polynomial. After finding one zero, say x = 1/5, we can factor the polynomial further.
Using synthetic division with x = 1/5:
By applying polynomial long division or synthetic division, we can find the other zeros of the reduced polynomial (a quadratic polynomial). Then we can solve the new quadratic for its zeros either by factoring or by using the quadratic formula:
x = (-b ± √(b² – 4ac))/(2a)
Once we find the zeros, we can determine the multiplicity of each zero. The multiplicity of a zero refers to the number of times that zero is a solution to the equation. For example, if a zero is derived from a factor of (x – r)², then that zero has a multiplicity of 2.
After you find all the zeros and their respective multiplicities, make sure to verify your answers by substituting them back into the original polynomial to confirm they equal zero.