Answer :
To construct a polynomial function [tex]\( f(x) \)[/tex] with the given properties, we will proceed through the following steps:
1. Identify the zeros of the polynomial: We are given that the polynomial has zeros at [tex]\( x = 1 \)[/tex], [tex]\( x = 2 \)[/tex], and [tex]\( x = 3 \)[/tex]. This means the polynomial has factors of the form [tex]\( (x - 1) \)[/tex], [tex]\( (x - 2) \)[/tex], and [tex]\( (x - 3) \)[/tex].
2. Form the polynomial with the identified zeros: A polynomial with these zeros can be written as:
[tex]\[ f(x) = k (x - 1)(x - 2)(x - 3) \][/tex]
where [tex]\( k \)[/tex] is a constant.
3. Ensure [tex]\( f(x) < 0 \)[/tex] for all values of [tex]\( x \)[/tex]: To make sure that the polynomial is negative for all [tex]\( x \)[/tex], we can choose a negative value for [tex]\( k \)[/tex]. Let's pick [tex]\( k = -1 \)[/tex].
4. Construct the polynomial: Substituting [tex]\( k = -1 \)[/tex] into the polynomial, we have:
[tex]\[ f(x) = - (x - 1)(x - 2)(x - 3) \][/tex]
5. Expand the polynomial: Now we'll expand the polynomial:
[tex]\[ f(x) = - (x - 1)(x - 2)(x - 3) \][/tex]
First, expand the product of two factors:
[tex]\[ (x - 1)(x - 2) = x^2 - 2x - x + 2 = x^2 - 3x + 2 \][/tex]
Next, multiply this result by [tex]\( (x - 3) \)[/tex]:
[tex]\[ f(x) = - \left( (x^2 - 3x + 2)(x - 3) \right) \][/tex]
Continue expanding:
[tex]\[ = - \left( x^3 - 3x^2 + 2x - 3x^2 + 9x - 6 \right) \][/tex]
Combine like terms inside the parentheses:
[tex]\[ = - (x^3 - 6x^2 + 11x - 6) \][/tex]
Finally, distribute the negative sign:
[tex]\[ f(x) = -x^3 + 6x^2 - 11x + 6 \][/tex]
So, the polynomial function [tex]\( f(x) \)[/tex] that meets all the requirements is:
[tex]\[ f(x) = -x^3 + 6x^2 - 11x + 6 \][/tex]
### Verification
1. Zeros at 1, 2, and 3:
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = -1^3 + 6(1)^2 - 11(1) + 6 = -1 + 6 - 11 + 6 = 0 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = -2^3 + 6(2)^2 - 11(2) + 6 = -8 + 24 - 22 + 6 = 0 \][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = -3^3 + 6(3)^2 - 11(3) + 6 = -27 + 54 - 33 + 6 = 0 \][/tex]
2. [tex]\( f(x) < 0 \)[/tex] for all values of [tex]\( x \)[/tex]:
- By choosing [tex]\( k = -1 \)[/tex], we ensured the polynomial will always be negative because the product [tex]\( (x - 1)(x - 2)(x - 3) \)[/tex] will always be positive for any real value of [tex]\( x \)[/tex] due to the odd number of factors crossed when [tex]\( x \)[/tex] is crossed zero points. Multiplying by [tex]\(-1\)[/tex] flips the sign making the function negative for all [tex]\( x \)[/tex].
3. Degree greater than 1:
- The resulting polynomial is of degree 3, which is greater than 1.
Therefore, the polynomial function satisfies all the given properties.
1. Identify the zeros of the polynomial: We are given that the polynomial has zeros at [tex]\( x = 1 \)[/tex], [tex]\( x = 2 \)[/tex], and [tex]\( x = 3 \)[/tex]. This means the polynomial has factors of the form [tex]\( (x - 1) \)[/tex], [tex]\( (x - 2) \)[/tex], and [tex]\( (x - 3) \)[/tex].
2. Form the polynomial with the identified zeros: A polynomial with these zeros can be written as:
[tex]\[ f(x) = k (x - 1)(x - 2)(x - 3) \][/tex]
where [tex]\( k \)[/tex] is a constant.
3. Ensure [tex]\( f(x) < 0 \)[/tex] for all values of [tex]\( x \)[/tex]: To make sure that the polynomial is negative for all [tex]\( x \)[/tex], we can choose a negative value for [tex]\( k \)[/tex]. Let's pick [tex]\( k = -1 \)[/tex].
4. Construct the polynomial: Substituting [tex]\( k = -1 \)[/tex] into the polynomial, we have:
[tex]\[ f(x) = - (x - 1)(x - 2)(x - 3) \][/tex]
5. Expand the polynomial: Now we'll expand the polynomial:
[tex]\[ f(x) = - (x - 1)(x - 2)(x - 3) \][/tex]
First, expand the product of two factors:
[tex]\[ (x - 1)(x - 2) = x^2 - 2x - x + 2 = x^2 - 3x + 2 \][/tex]
Next, multiply this result by [tex]\( (x - 3) \)[/tex]:
[tex]\[ f(x) = - \left( (x^2 - 3x + 2)(x - 3) \right) \][/tex]
Continue expanding:
[tex]\[ = - \left( x^3 - 3x^2 + 2x - 3x^2 + 9x - 6 \right) \][/tex]
Combine like terms inside the parentheses:
[tex]\[ = - (x^3 - 6x^2 + 11x - 6) \][/tex]
Finally, distribute the negative sign:
[tex]\[ f(x) = -x^3 + 6x^2 - 11x + 6 \][/tex]
So, the polynomial function [tex]\( f(x) \)[/tex] that meets all the requirements is:
[tex]\[ f(x) = -x^3 + 6x^2 - 11x + 6 \][/tex]
### Verification
1. Zeros at 1, 2, and 3:
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = -1^3 + 6(1)^2 - 11(1) + 6 = -1 + 6 - 11 + 6 = 0 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = -2^3 + 6(2)^2 - 11(2) + 6 = -8 + 24 - 22 + 6 = 0 \][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = -3^3 + 6(3)^2 - 11(3) + 6 = -27 + 54 - 33 + 6 = 0 \][/tex]
2. [tex]\( f(x) < 0 \)[/tex] for all values of [tex]\( x \)[/tex]:
- By choosing [tex]\( k = -1 \)[/tex], we ensured the polynomial will always be negative because the product [tex]\( (x - 1)(x - 2)(x - 3) \)[/tex] will always be positive for any real value of [tex]\( x \)[/tex] due to the odd number of factors crossed when [tex]\( x \)[/tex] is crossed zero points. Multiplying by [tex]\(-1\)[/tex] flips the sign making the function negative for all [tex]\( x \)[/tex].
3. Degree greater than 1:
- The resulting polynomial is of degree 3, which is greater than 1.
Therefore, the polynomial function satisfies all the given properties.
