Question

For each polynomial function, (a) list all possible rational zeros, (b) use a graph to eliminate some of the possible zeros listed in part (a), (c) find all rational zer…

For each polynomial function, (a) list all possible rational zeros, (b) use a graph to eliminate some of the possible zeros listed in part (a), (c) find all rational zeros, and (d) factor P(x). P(X)=x^3-2x^2-13x-10

Answer 1

(a) ±{1, 2, 5, 10}

(b) attached

(c) {-2, -1, 5}

(d) P(x) = (x +2)(x +1)(x -5)

Explanation:

You want the possible and actual rational zeros for P(x) = x^3-2x^2-13x-10, and the factored form of the polynomial.

(a) Rational zeros

The rational root theorem tells you the possible rational zeros of a polynomial will be of the form ...

±(divisor of the constant)/(divisor of the leading coefficient)

The leading coefficient is 1, and the divisors of the constant are {1, 2, 5, 10}, so the possible rational zeros are ...

{±1, ±2, ±5, ±10}

(b, c) Actual zeros

The attached graph shows the actual zeros of P(x) to be {-2, -1, 5}.

(d) Factors

Each zero q corresponds to a factor (x -q). The factored form of the polynomial is ...

P(x) = (x +2)(x +1)(x -5)