Math  /  Algebra

QuestionFind the zeros and multiplicity of the function f(x)=5x330x2+45x f(x) = 5x^3 - 30x^2 + 45x .

Studdy Solution

STEP 1

1. We are given the polynomial function f(x)=5x330x2+45x f(x) = 5x^3 - 30x^2 + 45x .
2. We need to find the zeros of the function.
3. We need to determine the multiplicity of each zero.

STEP 2

1. Factor the polynomial completely.
2. Identify the zeros from the factored form.
3. Determine the multiplicity of each zero.

STEP 3

Factor out the greatest common factor (GCF) from the polynomial:
f(x)=5x(x26x+9) f(x) = 5x(x^2 - 6x + 9)

STEP 4

Factor the quadratic expression x26x+9 x^2 - 6x + 9 :
Notice that x26x+9 x^2 - 6x + 9 is a perfect square trinomial:
x26x+9=(x3)2 x^2 - 6x + 9 = (x - 3)^2
Thus, the factored form of the polynomial is:
f(x)=5x(x3)2 f(x) = 5x(x - 3)^2

STEP 5

Identify the zeros from the factored form:
Set each factor equal to zero:
1. 5x=0 5x = 0 gives x=0 x = 0 .
2. (x3)2=0 (x - 3)^2 = 0 gives x=3 x = 3 .

STEP 6

Determine the multiplicity of each zero:
1. The zero x=0 x = 0 comes from the factor 5x 5x , which has a multiplicity of 1.
2. The zero x=3 x = 3 comes from the factor (x3)2 (x - 3)^2 , which has a multiplicity of 2.

The zeros of the function are x=0 x = 0 with multiplicity 1, and x=3 x = 3 with multiplicity 2.

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