Math  /  Algebra

QuestionFind the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the x -axis or touches the x-axis and turns around at each zero. f(x)=x3+3x24x12f(x)=x^{3}+3 x^{2}-4 x-12
Determine the zero(s), if they exist. The zero(s) is/are \square . (Type integers or decimals. Use a comma to separate answers as needed.)

Studdy Solution

STEP 1

1. We are given the polynomial function f(x)=x3+3x24x12 f(x) = x^3 + 3x^2 - 4x - 12 .
2. We need to find the zeros of the polynomial.
3. We need to determine the multiplicity of each zero.
4. We need to state whether the graph crosses or touches the x-axis at each zero.

STEP 2

1. Use the Rational Root Theorem to identify possible rational zeros.
2. Perform synthetic division or polynomial division to test possible zeros.
3. Factor the polynomial completely.
4. Determine the zeros and their multiplicities.
5. Analyze the behavior of the graph at each zero.

STEP 3

Use the Rational Root Theorem. The possible rational zeros are the factors of the constant term (-12) divided by the factors of the leading coefficient (1). Therefore, the possible rational zeros are:
±1,±2,±3,±4,±6,±12 \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12

STEP 4

Test these possible zeros using synthetic division. Start with x=2 x = 2 :
Perform synthetic division with x=2 x = 2 :
213412210121560\begin{array}{r|rrrr} 2 & 1 & 3 & -4 & -12 \\ & & 2 & 10 & 12 \\ \hline & 1 & 5 & 6 & 0 \\ \end{array}
Since the remainder is 0, x=2 x = 2 is a zero.

STEP 5

The quotient from the synthetic division is x2+5x+6 x^2 + 5x + 6 . Factor this quadratic:
x2+5x+6=(x+2)(x+3) x^2 + 5x + 6 = (x + 2)(x + 3)
Thus, the complete factorization of the polynomial is:
f(x)=(x2)(x+2)(x+3) f(x) = (x - 2)(x + 2)(x + 3)

STEP 6

The zeros of the polynomial are x=2,x=2, x = 2, x = -2, and x=3 x = -3 .
Each zero has a multiplicity of 1.

STEP 7

Analyze the behavior at each zero:
- At x=2 x = 2 , the graph crosses the x-axis because the multiplicity is odd. - At x=2 x = -2 , the graph crosses the x-axis because the multiplicity is odd. - At x=3 x = -3 , the graph crosses the x-axis because the multiplicity is odd.
The zero(s) is/are 2,2,3 2, -2, -3 .

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