Find all the real zeros of the polynomial. Use the quadratic formula if necessary.
4x^3 − 10x^2 + 9
5/4 +/- 2 i sqrt(11)
To find the real zeros of the polynomial 4x^3 - 10x^2 + 9, we need to solve the equation 4x^3 - 10x^2 + 9 = 0.
Unfortunately, this polynomial is not quadratic, so we cannot directly apply the quadratic formula. However, we can still solve it by factoring if possible or by using numerical methods such as graphing or synthetic division.
Let's try factoring the polynomial.
Step 1: Check for any common factors among the coefficients.
The coefficients 4, -10, and 9 have no common factors other than 1, so we cannot factor out any constants here.
Step 2: Check for any common factors among the terms.
The terms 4x^3, -10x^2, and 9 do not share any common factors, so we cannot factor out any variables.
Step 3: Use the Rational Root Theorem to find possible rational zeros.
The Rational Root Theorem states that if a polynomial has any rational zeros, they must be of the form p/q, where p is a factor of the constant term (9 in this case) and q is a factor of the leading coefficient (4 in this case).
The factors of 9 are ±1, ±3, and ±9, and the factors of 4 are ±1 and ±2. So the possible rational zeros are ±1/1, ±3/1, ±9/1, ±1/2, ±3/2, and ±9/2.
Step 4: Test the possible zeros using synthetic division or substitute them into the polynomial equation.
By testing the possible rational zeros, we find that none of them yield a remainder of 0 when substituted into the polynomial equation 4x^3 - 10x^2 + 9 = 0.
Since factoring did not lead to any real zeros and substitution did not yield any rational zeros, we can conclude that the polynomial 4x^3 - 10x^2 + 9 does not have any rational zeros. At this point, we can resort to numerical methods, such as graphing the polynomial or using iterative methods to find approximate solutions.
Please note that finding the exact real zeros of cubic polynomials can be challenging and might require advanced techniques like Cardano's method or calculus-based methods.