To find a polynomial function of degree 4 with the given zeros and their respective multiplicities, and which passes through a specific point, follow these steps:
### Step 1: Identify the Zeros and Construct the Polynomial
The given zeros are:
- [tex]\( -2 \)[/tex] with a multiplicity of 2
- [tex]\( 2 \)[/tex] with a multiplicity of 2
Since these zeros have multiplicities greater than 1, they can be represented as repeated factors in the polynomial. Therefore, our polynomial can be written as:
[tex]\[ P(x) = a(x + 2)^2(x - 2)^2 \][/tex]
where [tex]\( a \)[/tex] is a leading coefficient that we need to determine.
### Step 2: Use the Given Point to Find the Leading Coefficient
We are given that the polynomial passes through the point [tex]\( (-3, 50) \)[/tex]. This means when [tex]\( x = -3 \)[/tex], the value of [tex]\( P(x) \)[/tex] should be 50. Thus, we substitute [tex]\( x = -3 \)[/tex] and [tex]\( P(x) = 50 \)[/tex] into our polynomial equation:
[tex]\[ 50 = a(-3 + 2)^2(-3 - 2)^2 \][/tex]
Simplify the equation:
[tex]\[ 50 = a(-1)^2(-5)^2 \][/tex]
[tex]\[ 50 = a \cdot 1 \cdot 25 \][/tex]
[tex]\[ 50 = 25a \][/tex]
Solve for [tex]\( a \)[/tex]:
[tex]\[ a = \frac{50}{25} \][/tex]
[tex]\[ a = 2 \][/tex]
### Step 3: Construct the Final Polynomial
Now that we have the value of the leading coefficient [tex]\( a \)[/tex], we can write the final polynomial:
[tex]\[ P(x) = 2(x + 2)^2(x - 2)^2 \][/tex]
### Step 4: Expand the Polynomial (Optional)
To provide the polynomial in its expanded form, we expand [tex]\( (x + 2)^2 \)[/tex] and [tex]\( (x - 2)^2 \)[/tex] first:
[tex]\[ (x + 2)^2 = x^2 + 4x + 4 \][/tex]
[tex]\[ (x - 2)^2 = x^2 - 4x + 4 \][/tex]
Then multiply these results together:
[tex]\[ (x^2 + 4x + 4)(x^2 - 4x + 4) \][/tex]
Using the distributive property (FOIL method):
[tex]\[ = x^2(x^2 - 4x + 4) + 4x(x^2 - 4x + 4) + 4(x^2 - 4x + 4) \][/tex]
[tex]\[ = x^4 - 4x^3 + 4x^2 + 4x^3 - 16x^2 + 16x + 4x^2 - 16x + 16 \][/tex]
[tex]\[ = x^4 + (0)x^3 + (4x^2 - 16x^2 + 4x^2) + (16x - 16x) + 16 \][/tex]
[tex]\[ = x^4 - 8x^2 + 16 \][/tex]
Now multiply by 2:
[tex]\[ 2(x^4 - 8x^2 + 16) \][/tex]
[tex]\[ = 2x^4 - 16x^2 + 32 \][/tex]
Therefore, the polynomial in expanded form is:
[tex]\[ P(x) = 2x^4 - 16x^2 + 32 \][/tex]
### Final Answer
The polynomial function of degree 4 with the given properties is:
[tex]\[ P(x) = 2(x + 2)^2(x - 2)^2 \][/tex]
In expanded form, it is:
[tex]\[ P(x) = 2x^4 - 16x^2 + 32 \][/tex]
