109k views
1 vote
Given one zero of the polynomial function, find the other zeros.
f(x)=x^3+2x^2-20x+24; -6

User RaphaelH
by
8.6k points

1 Answer

6 votes

Answer:

{2, 2, -6}

Explanation:

synthetic division is one of the easier ways of determining whether or not a given number is a root of a polynomial. Here we're told that -6 is a root. Let's go through the steps of synthetic division: keep in mind that if there is no remainder (that is, the remainder is zero), then the divisor (such as -6) is a zero of the polynomial.

The coefficients of f(x)=x^3+2x^2-20x+24 are {1, 2, -20, 24}.

Setting up synthetic div.:

-6 ) 1 2 -20 24

-6 24 -24

----------------------------

1 -4 4 0

Here the remainder is zero (0), so it is safe to assume that -6 is a zero of the original polynomial.

We can continue with synth. div. to determine the remaining zeros of the original function f(x)=x^3+2x^2-20x+24. The coefficients {1, -4, 4} represent the quadratic x^2 - 4x + 4. Let's check whether the factor 2 of 4 is indeed a zero:

2 ) 1 -4 4

2 -4

-----------------

1 -2 0

Here the remainder is zero. Thus, 2 is a zero of f(x)=x^3+2x^2-20x+24. Notice that the remaining coefficients are {1, -2}; they represent x - 2 = 0, so the third root is 2.

Thus, the zeros of f(x)=x^3+2x^2-20x+24 are {2, 2, -6}

User Sebastian Otto
by
8.2k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories