Real Zeros of Polynomial Function...1

[nycmathguy]

Section 2.2
Question 36

Can you solve 36 parts (a-d) as a guide for me to do a few more?

 

[MathLover1]

36.
f(x)=x^2+10x+25...........recognize the square of the sum here (a+b)^2=a^2+2ab+b^2

f(x)=x^2+10x+5^2

f(x)=(x+5)^2

a.
real zeros:
(x+5)^2=0
(x+5)(x+5)=0 (note we will have same zero twice)
x+5=0=>x =-5=> multiplicity 2

b.

If the graph touches the x-axis and bounces off of the axis, it is a zero with even multiplicity.
If the graph crosses the x-axis at a zero, it is a zero with odd multiplicity.

in this case, it is a zero with even multiplicity

c.
The maximum number of turning points of a polynomial function is always one less than the degree of the function.

in this case the degree of the function is 2, so there is 1 turning point

d.

 

[nycmathguy]

36.
f(x)=x^2+10x+25...........recognize the square of the sum here (a+b)^2=a^2+2ab+b^2

f(x)=x^2+10x+5^2

f(x)=(x+5)^2

a.
real zeros:
(x+5)^2=0
(x+5)(x+5)=0 (note we will have same zero twice)
x+5=0=>x =-5=> multiplicity 2

b.

If the graph touches the x-axis and bounces off of the axis, it is a zero with even multiplicity.
If the graph crosses the x-axis at a zero, it is a zero with odd multiplicity.

in this case, it is a zero with even multiplicity

c.
The maximum number of turning points of a polynomial function is always one less than the degree of the function.

in this case the degree of the function is 2, so there is 1 turning point

d.

You said:

(note we will have same zero twice)
x+5=0=>x =-5=> multiplicity 2

What is multiplicity?
Why is the multiplicity 2?

You said:

If the graph touches the x-axis and bounces off of the axis, it is a zero with even multiplicity.

What do you mean it is zero with even multiplicity?
Why not odd multiplicity?

You said:

If the graph crosses the x-axis at a zero, it is a zero with odd multiplicity.

Can you explain this multiplicity stuff another way? I'm confused. Sorry.

You said:

The maximum number of turning points of a polynomial function is always one less than the degree of the function.

Can you please use a graph to explain your statement?

You said:

in this case the degree of the function is 2, so there is 1 turning point

Why only a single turning point? This is not making sense to me. Sorry.

Question:

Can you find a good video on You Tube that explains this material? Sometimes, I learn certain topics better watching a video lesson.

Thanks

 

[MathLover1]

A zero has a "multiplicity", which refers to the number of times that its associated factor appears in the polynomial. The multiplicity of each zero is the number of times that its corresponding factor appears. In other words, the multiplicities are the powers. (For the factor x-5, the understood power is 1.)


for example, if the polynomial function is:

y = 3(x + 5)^3 *(x + 2)^4* (x – 1)^2* (x – 5)

than zeros 3(x + 5)^3 *(x + 2)^4* (x – 1)^2* (x – 5) =0
each factor equal to zero, and you get

x = –5 with multiplicity 3 (because (x + 5)^3 is (x + 5) to power of 3)
x = –2 with multiplicity 4
x = 1 with multiplicity 2
x = 5 with multiplicity 1

(x+5)^2=0 (note power is 2)
x+5=0
=>x =-5=> multiplicity 2

turning point: is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising)

as you can see a parabola (no matter which one, one that opens down or one that opens up) has one turning point

 

[nycmathguy]

A zero has a "multiplicity", which refers to the number of times that its associated factor appears in the polynomial. The multiplicity of each zero is the number of times that its corresponding factor appears. In other words, the multiplicities are the powers. (For the factor x-5, the understood power is 1.)


for example, if the polynomial function is:

y = 3(x + 5)^3 *(x + 2)^4* (x – 1)^2* (x – 5)

than zeros 3(x + 5)^3 *(x + 2)^4* (x – 1)^2* (x – 5) =0
each factor equal to zero, and you get

x = –5 with multiplicity 3 (because (x + 5)^3 is (x + 5) to power of 3)
x = –2 with multiplicity 4
x = 1 with multiplicity 2
x = 5 with multiplicity 1

(x+5)^2=0 (note power is 2)
x+5=0
=>x =-5=> multiplicity 2

turning point: is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising)
View attachment 334

View attachment 335

as you can see a parabola (no matter which one, one that opens down or one that opens up) has one turning point

Wow! You went above and beyond here. I thank you. I owe you.