Question

What are the zeros of the function? What are their multiplicities? y=(x-2)(x+7)^3

Answer 1

Answer:

1)$S=\left \{ 2,-7 \right \}$ 2) x=2, multiplicity 1 x=-7, multiplicity is 3.

Step-by-step explanation:

1) The zeros of this function are the roots of it. Which number added to -2 will return 0 for the first factor (x-2)? Similarly, which one added to -7 for the second factor (x+7)³ will return 0?

$y=(x-2)(x+7)^3\Rightarrow (x-2)(x+7)^3=0\rightarrow (x-2)=0\\\therefore(2-2)(x+7)^{3}=0\Rightarrow 0(x+7)^{3}=0\,x'=2\\(x-2)(-7+7)^{3}=0\Rightarrow (x-2)(0)^{3}=0\.x''=-7\\S=\left \{ 2,-7 \right \}$

2) The multiplicity of a function is the number of repeated times, a factor  of a polynomial function appears in its factored form. So,

$y=(x-2)(x+7)^3\Rightarrow y=(x-2)(x+7)(x+7)(x+7)$

The factor is (x+7) whose multiplicity is 3, then x=-7, multiplicity=3

The factor (x-2), then x=2, multiplicity =1

Answer 2

(2,0) is a single
(-7,0) is a triple