Question

week 7
find all zeros of this polynomial (f(x)=x^{3}+13x^{2}+47x + 35) if (x = - 1) is a zero. what are the rest?
zeros: (x=-1,x=), (x=)

Explanation:

Step1: Use synthetic division

Since \(x = - 1\) is a zero of \(f(x)=x^{3}+13x^{2}+47x + 35\), we perform synthetic - division. The coefficients are \(1,13,47,35\).

-1 |  1  13  47  35
    |     -1 -12 -35
    |________________
      1  12  35   0

The quotient is \(x^{2}+12x + 35\).

Step2: Factor the quotient

We factor the quadratic \(x^{2}+12x + 35\). We need to find two numbers that multiply to \(35\) and add up to \(12\). The numbers are \(5\) and \(7\). So \(x^{2}+12x + 35=(x + 5)(x+7)\).

Step3: Find the remaining zeros

Set \((x + 5)(x + 7)=0\). Using the zero - product property, if \(x+5 = 0\), then \(x=-5\); if \(x + 7=0\), then \(x=-7\).

Answer:

\(x=-5,x=-7\)