Exercise 13 Page 965

We want to find the number of distinct real roots for the given equation. We will start by factoring out the greatest common factor, x. x^4 +3x^3-4x = 0 ⇕ x(x^3+3x^2-4) = 0 Now, let's recall the Zero-Product Property. If ab=0, then a=0 or b=0This means that we can divide the given equation into two polynomials. x = 0 x^3+3x^2-4= 0 The first polynomial indicates that 0 is a root of the given equation. The remaining roots can be found by solving the second polynomial. To do it, let's start by rewriting the squared term as the sum of two terms so that it is easier to factor by grouping.

Notice that now we can factor out (x+2). Then, we will rewrite the linear term as a difference of two terms to continue factoring.

Finally, we will apply the Zero-Product Property to find the roots.

Now that we have the roots for both polynomials, we can write all the real roots of the original equation. x_1 = 0, x_2= 1 , x_3=- 2 , x_4=- 2 Note that - 2 is a double root, so the given equation has only three distinct real roots. This corresponds to option C.