Let's go through the steps required to write the equation of a cubic polynomial function given its roots (zeroes) and a point it passes through:
### Step 1: Express the General Form of the Cubic Polynomial
First, recognize that a cubic polynomial with roots [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] can be written as:
[tex]\[ f(x) = k(x - a)(x - b)(x - c) \][/tex]
where [tex]\( k \)[/tex] is a constant coefficient.
### Step 2: Use the Information from a Given Point
We know that the polynomial passes through the point [tex]\((0, -5)\)[/tex]. This means that when [tex]\( x = 0 \)[/tex], [tex]\( f(x) = -5 \)[/tex]. Substitute these values into the polynomial to solve for [tex]\( k \)[/tex]:
[tex]\[ f(0) = k(0 - a)(0 - b)(0 - c) \][/tex]
[tex]\[ -5 = k(-a)(-b)(-c) \][/tex]
[tex]\[ -5 = -kab \cdot c \][/tex]
[tex]\[ -5 = -kabc \][/tex]
### Step 3: Solve for the Constant [tex]\( k \)[/tex]
To isolate [tex]\( k \)[/tex], divide both sides by [tex]\(-abc\)[/tex]:
[tex]\[ k = \frac{5}{abc} \][/tex]
### Step 4: Substitute [tex]\( k \)[/tex] Back into the Polynomial
Now that we know [tex]\( k \)[/tex], substitute it back into the polynomial:
[tex]\[ f(x) = \frac{5}{abc}(x - a)(x - b)(x - c) \][/tex]
### Step 5: Write the Complete Equation
Thus, the complete equation for our cubic polynomial, given the roots [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex], is:
[tex]\[ f(x) = \frac{5}{abc}(x - a)(x - b)(x - c) \][/tex]
### Example with Specific Roots
Suppose our roots are [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = 3 \)[/tex]. Substituting these values in, we get:
[tex]\[ abc = 1 \cdot 2 \cdot 3 = 6 \][/tex]
[tex]\[ k = \frac{5}{6} \][/tex]
[tex]\[ f(x) = \frac{5}{6}(x - 1)(x - 2)(x - 3) \][/tex]
So, the specific polynomial function that passes through the point [tex]\((0,-5)\)[/tex] and has roots [tex]\( x = 1, 2, 3 \)[/tex] is:
[tex]\[ f(x) = \frac{5}{6}(x - 1)(x - 2)(x - 3) \][/tex]
This final expression represents the cubic polynomial function that meets the given conditions.
