Questions: Find all rational zeros of the polynomial. (Enter your answers a P(x)=3 x^5-20 x^4-16 x^3+54 x^2+61 x+14 x=-1,1,2,-1/3, 7x Write the polynomial in factored form. P(x)=(x+1)^2(3 x+1)(x-2)(x-7)

Find all rational zeros of the polynomial. (Enter your answers a

P(x)=3 x^5-20 x^4-16 x^3+54 x^2+61 x+14
x=-1,1,2,-1/3, 7x

Write the polynomial in factored form.

P(x)=(x+1)^2(3 x+1)(x-2)(x-7)

Transcript text: Find all rational zeros of the polynomial. (Enter your answers a \[ \begin{array}{l} P(x)=3 x^{5}-20 x^{4}-16 x^{3}+54 x^{2}+61 x+14 \\ x=-1,1,2,-\frac{1}{3}, 7{ }_{x} \end{array} \] Write the polynomial in factored form. \[ P(x)=(x+1)^{2}(3 x+1)(x-2)(x-7) \]

Solution

Solution Steps

Solution Approach

Verify the given rational zeros by substituting them into the polynomial \( P(x) \) and checking if they yield zero.
Use the Rational Root Theorem to list all possible rational zeros.
Use synthetic division or polynomial division to factorize the polynomial using the verified zeros.
Write the polynomial in its factored form using the identified zeros.

Step 1: Verify Rational Zeros

We are given the polynomial

\[ P(x) = 3x^5 - 20x^4 - 16x^3 + 54x^2 + 61x + 14 \]

and the potential rational zeros \( -1, 1, 2, -\frac{1}{3}, 7 \). By substituting these values into \( P(x) \), we find that the verified rational zeros are \( -1, 2, \) and \( 7 \).

Step 2: Factor the Polynomial

Using the verified rational zeros, we can factor the polynomial. The factored form of \( P(x) \) is given by:

\[ P(x) = (x - 7)(x - 2)(x + 1)^2(3x + 1) \]