Answer :
Sure, let's find the polynomial function of least degree with a leading coefficient of 1, which can be expressed as:
[tex]\[ f(x) = x^3 + A x^2 + B x + C \][/tex]
### Step-by-Step Solution
1. Express the General Form:
We are given the form of the polynomial:
[tex]\[ f(x) = x^3 + A x^2 + B x + C \][/tex]
2. Determine the Coefficients:
- The coefficient of [tex]\(x^3\)[/tex] is 1, which is the leading coefficient.
- We need to determine the coefficients [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex].
3. Coefficients Known:
The coefficients have been given as:
- [tex]\(A = 2\)[/tex]
- [tex]\(B = -4\)[/tex]
- [tex]\(C = 5\)[/tex]
4. Substitute the Coefficients into the Polynomial:
We substitute [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] into the polynomial formula:
[tex]\[ f(x) = x^3 + 2 x^2 - 4 x + 5 \][/tex]
Therefore, the polynomial function of least degree with a leading coefficient of 1 is:
[tex]\[ \boxed{f(x) = x^3 + 2 x^2 - 4 x + 5} \][/tex]
The coefficients are:
- [tex]\(A = 2\)[/tex]
- [tex]\(B = -4\)[/tex]
- [tex]\(C = 5\)[/tex]
These values give us the complete polynomial function.
[tex]\[ f(x) = x^3 + A x^2 + B x + C \][/tex]
### Step-by-Step Solution
1. Express the General Form:
We are given the form of the polynomial:
[tex]\[ f(x) = x^3 + A x^2 + B x + C \][/tex]
2. Determine the Coefficients:
- The coefficient of [tex]\(x^3\)[/tex] is 1, which is the leading coefficient.
- We need to determine the coefficients [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex].
3. Coefficients Known:
The coefficients have been given as:
- [tex]\(A = 2\)[/tex]
- [tex]\(B = -4\)[/tex]
- [tex]\(C = 5\)[/tex]
4. Substitute the Coefficients into the Polynomial:
We substitute [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] into the polynomial formula:
[tex]\[ f(x) = x^3 + 2 x^2 - 4 x + 5 \][/tex]
Therefore, the polynomial function of least degree with a leading coefficient of 1 is:
[tex]\[ \boxed{f(x) = x^3 + 2 x^2 - 4 x + 5} \][/tex]
The coefficients are:
- [tex]\(A = 2\)[/tex]
- [tex]\(B = -4\)[/tex]
- [tex]\(C = 5\)[/tex]
These values give us the complete polynomial function.
