Write the equation of the polynomial function with the following real zeros and multiplicity if (2,16) is a point on the graph
zero at x=-2 with a with a multiplicity of 2
zero at x=1 with a multiplicity of 3
zero at x=0 with a multiplicity of 1
I did x(x+2)^2 (x-1)^3
16 =a^2(2+2)^2 (2-1)^3
16=a^22(4)^2 (1)^3
16=a^2(16) 1
16=a32 a=1/2
final answer is y=1/2(x+2)^2(x-1)^3 x
Is this correct? Thank you for checking my work
I wrote part of it incorrect up above.
Use this work.
I did x(x+2)^2 (x-1)^3
16 =a(2)(2+2)^2 (2-1)^3
16=a(2) (4)^2 (1)^3
16=a(2)(16) 1
16=a32 a=1/2
final answer is y=1/2(x+2)^2(x-1)^3 x
Yes : ) This is correct : )
Thank you
Your opening function should have been
y = ax(x+2)^2 (x-1)^3
you went from
x(x+2)^2 (x-1)^3 to
16 =a(2)(2+2)^2 (2-1)^3
all of sudden resurrecting an equal sign and the "a".
Just a case of "bad mathematical form". Remember, Math is a precise language.
To determine if your work is correct, let's go through the steps one by one.
Given zeros and multiplicities:
1. Zero at x = -2 with a multiplicity of 2.
2. Zero at x = 1 with a multiplicity of 3.
3. Zero at x = 0 with a multiplicity of 1.
The equation of a polynomial function with all the given zeros and multiplicities can be written as:
f(x) = a(x - zero1)^multiplicity1 * (x - zero2)^multiplicity2 * (x - zero3)^multiplicity3
1. First, we consider the zero at x = -2 with a multiplicity of 2:
(x - (-2))^2
Simplifying, we get (x + 2)^2.
2. Next, we consider the zero at x = 1 with a multiplicity of 3:
(x - 1)^3
3. Lastly, we consider the zero at x = 0 with a multiplicity of 1:
(x - 0)^1
Simplifying, we get x.
Combining all the factors, the equation becomes:
f(x) = a(x + 2)^2(x - 1)^3x
Now, let's check the given point (2, 16) by substituting x = 2 and y = 16 into the equation:
16 = a(2 + 2)^2(2 - 1)^3*2
16 = a(4)^2(1)^3*2
16 = a(16)*2
16 = 32a
Dividing both sides by 32:
a = 16/32
a = 1/2
Therefore, the final equation of the polynomial function is:
f(x) = (1/2)(x + 2)^2(x - 1)^3x
Great job! Your work is correct, and the equation of the polynomial function with the given zeros and multiplicity is f(x) = (1/2)(x + 2)^2(x - 1)^3x.