Determine real zeros and multiplicity of x=5(3x+4)^3(x-4)^2

Question

For the polynomial determine the degree.list each real zero and its multiplicity . Determine whether the graph crosses or touches the x axis at each x intercept

 

Answer 1

I don't know if there was a typo, and if this is actually asking about f(x) = 5(3x+4)³(x-4)² instead, so I'll assume that it's meant to be f(x).

 

To find the degree of the polynomial, you need to find the x with the largest power. Here, you have an x within a cube, and an x within a square. These two x's will eventually be multiplied together, so you'll have: x³*x² = x³⁺² = x⁵. This is the x with the largest power, so your polynomial is of a degree of 5.

 

Each real zero is the number that would cause the entire equation to equal zero. In this case, if 3x + 4 were to equal zero, then the entire equation would equal zero. Same goes for if x-4 were to equal zero. So you set both equations equal to zero and solve for x:

 

3x + 4 = 0

3x = -4 <--- Subtract both sides by 4

x = -4/3 <--Divide both sides by 3

 

x - 4 = 0

x = 4 <--- Add 4 to both sides

 

So you have two real zeros: x = -4/3 , 4

 

The multiplicity of each zero is determined by the power associated with each. For example, 3x + 4 has a power of 3. The zero associated with this equation is -4/3. Therefore, the multiplicity of -4/3 is 3. Similarly, the power of x - 4 is 2, so the multiplicity of the zero x = 4 is 2.

 

Lastly, a graph will touch the graph if the multiplicity is even. It will cross the graph if the multiplicity is odd. Therefore, it touches the graph at x = 4 and crosses at x = -4/3.