Find the zeros of the polynomial function and state the multiplicity of each. f(x) = 5(x + 6)^2(x - 6)^3

Question

 need help solving this question^

Answer 1

Cara, the zeroes of a function is where it crosses the x-axis.  All along the x-axis, y = 0, so we need to set our function equal to 0 to find the zeroes.

 

 Then (x+6) must equal 0 and/or (x-6) must equal 0 to make the whole thing equal 0, right?  5(0)²(x-6)³ = 0, etc.

 

If x + 6 = 0, then x = -6.  Since that is squared, which would look like (x + 6)(x + 6), then the multiplicity of -6 is 2.

 

If x - 6 = 0, then x = 6.  Since that is cubed, the multiplicity is 3.

 

Hope this helps!

Answer 2

Cara,

 

Zeros are those values of x for which f(x) = 0 .  Thus solve:

5(x + 6)²(x - 6)³ =0 which simplifies to when divided by 5:  (x + 6)²(x - 6)³ = 0

 

The only way that the left side product of 2 factors can be zero is if either or both factors are zero. Hence,

 

when (x + 6) = 0, (x + 6)² = 0, and x = -6  two roots because of the power 2.

when (x - 6) = 0, (x - 6)³ = 0,   and x = +6  three roots because of the power 3.