Question

Write the equation of the polynomial shown below in standard form.
(1,0), (2,4), (4,0), (5,4)

Answer 1

To find the polynomial that passes through the points (1,0), (2,4), (4,0), and (5,4), we start by identifying the roots and the overall structure of the polynomial.

### Step 1: Identify roots
The roots of the polynomial are given by the points where the polynomial is equal to zero, which are the x-coordinates of the points (1,0) and (4,0). Thus, the roots are:
- \( x = 1 \)
- \( x = 4 \)

### Step 2: Form factors from roots
Since these are roots, we can create factors from them:
- For root at \( x = 1 \): \( (x - 1) \)
- For root at \( x = 4 \): \( (x - 4) \)

### Step 3: Form polynomial with these factors
The polynomial can be expressed as:
\[
P(x) = k(x - 1)(x - 4)
\]
where \( k \) is a constant to be determined.

### Step 4: Using the remaining points to find k
We can use either of the points (2, 4) or (5, 4) to find \( k \). Let's use the point (2, 4):

Substituting \( x = 2 \) and \( P(x) = 4 \):
\[
4 = k(2 - 1)(2 - 4)
\]
\[
4 = k(1)(-2)
\]
\[
4 = -2k
\]
\[
k = -2
\]

### Step 5: Write the polynomial
Now substitute \( k \) back into the polynomial:
\[
P(x) = -2(x - 1)(x - 4)
\]

### Step 6: Expand to standard form
Now we expand \( P(x) \):
\[
P(x) = -2[(x^2 - 5x + 4)]
\]
\[
P(x) = -2x^2 + 10x - 8
\]

### Conclusion
Thus, the polynomial in standard form is:
\[
P(x) = -2x^2 + 10x - 8
\]