Find all the zeros of the polynomial function. Use the Rational Zero Theorem, Descartes's Rule of Signs obtaining the first zero. \( f(x)=x^{3}-4 x^{2}-7 x+10 \) The zeros of the function are \( \square \). (Use a comma to separate your answers as needed.)

To find the zeros of the polynomial function \( f(x) = x^3 - 4x^2 - 7x + 10 \), we will use the Rational Root Theorem and Descartes' Rule of Signs. ### Step 1: Apply Descartes' Rule of Signs First, we will analyze the sign changes in \( f(x) \): - The coefficients of \( f(x) \) are: \( 1, -4, -7, 10 \). - The signs are: \( +, -, -, + \). Counting the sign changes: - From \( +1 \) to \( -4 \): 1 change - From \( -4 \) to \( -7 \): 0 changes - From \( -7 \) to \( +10 \): 1 change Total sign changes = 2. According to Descartes' Rule of Signs, the number of positive real roots is either equal to the number of sign changes (2) or less than that by an even number. Therefore, there can be 2 or 0 positive real roots. Next, we will check for negative roots by evaluating \( f(-x) \): \[ f(-x) = (-x)^3 - 4(-x)^2 - 7(-x) + 10 = -x^3 - 4x^2 + 7x + 10 \] The coefficients of \( f(-x) \) are: \( -1, -4, 7, 10 \). The signs are: \( -, -, +, + \). Counting the sign changes: - From \( -1 \) to \( -4 \): 0 changes - From \( -4 \) to \( +7 \): 1 change - From \( +7 \) to \( +10 \): 0 changes Total sign changes = 1. Thus, there is 1 negative real root. ### Step 2: Use the Rational Root Theorem The Rational Root Theorem states that any rational root, in the form \( \frac{p}{q} \), must have \( p \) as a factor of the constant term (10) and \( q \) as a factor of the leading coefficient (1). Factors of 10: \( \pm 1, \pm 2, \pm 5, \pm 10 \) Possible rational roots: \( \pm 1, \pm 2, \pm 5, \pm 10 \) ### Step 3: Test Possible Rational Roots We will test these possible rational roots in \( f(x) \): 1. **Testing \( x = 1 \)**: \[ f(1) = 1^3 - 4(1^2) - 7(1) + 10 = 1 - 4 - 7 + 10 = 0 \] So, \( x = 1 \) is a root. ### Step 4: Factor the Polynomial Now that we have found one root, we can factor \( f(x) \) using synthetic division by \( x - 1 \): \[ \begin{array}{r|rrrr} 1 & 1 & -4 & -7 & 10 \\ & & 1 & -3 & -10 \\ \hline & 1 & -3 & -10 & 0 \\ \end{array} \] The result is \( x^2 - 3x - 10 \). ### Step 5: Factor the Quadratic Now we need to factor \( x^2 - 3x - 10 \): \[ x^2 - 3x - 10 = (x - 5)(x + 2) \] ### Step 6: Find All Zeros Thus, the complete factorization of \( f(x) \) is: \[ f(x) = (x - 1)(x - 5)(x + 2) \] The zeros of the function are: \[ x = 1, x = 5, x = -2 \] ### Final Answer The zeros of the function are \( 1, 5, -2 \).