Question

Obtain all zeros of f(x) = x³ + 13x² + 32x + 20, if one of its zeros is –2.

Answer 1

Since –2 is one zero of f(x)

Therefore, we know that, if x = a is a zero of a polynomial, then x – a is a factor of f(x) = x + 2 is a factor of f(x)

Now, we divide f(x) = x³ + 13x² + 32x + 20 by g(x) = (x + 2) to find the others zeros of f(x).

               x² – 11x + 10
`x + 2")"overline(+cancel(x^3) + 13x^2 + 32x + 20)`
           `+ \cancel(x^3) + 2x^2`
            –        –                              
                     `+ \cancel(11x^2) + 32x`
                     `+ \cancel(11x^2) + 22x`
                      –            –                     
                                   `+ \cancel(10x) + \cancel(20)`
                                   `+ \cancel(10x) + \cancel(20)`
                                    –         –          
                                               0

By using that division algorithm we have,

f(x) = g(x) x q(x) + r(x)

x³ + 13x² + 32x + 20 = (x + 2)(x² + 11x + 10) + 0

x³ + 13x² + 32x + 20 = (x + 2)(x² + 10x + 1x + 10)

x³ + 13x² + 32x + 20 = (x + 2)[x(x + 10) + 1(x + 10)]

x³ + 13x² + 32x + 20 = (x + 2)(x + 1)(x + 10)

Hence, the zeros of the given polynomials are –2, –1, and –10.