Find the remainder when the polynomial is divided by x-4

To find the remainder when the polynomial \( f(x) \) is divided by \( x-4 \), we need to evaluate \( f(4) \) using the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is \( f(c) \).

Since we don't have the explicit polynomial function \( f(x) \) from the graph, we can attempt to form an equation for the polynomial by using the given points.

From the graph, we can infer that the polynomial \( f(x) \) is cubic, meaning it has the form:
\[ f(x) = ax^3 + bx^2 + cx + d \]

We can use the points:
1. \( (-2, 0) \)
2. \( (-1, 0) \)
3. \( (2, 0) \)
4. A point such as \( B = (0.319, 8.643) \) to set up the system of equations and solve for the coefficients \( a, b, c, \) and \( d \).

Using the roots and their multiplicity, we have:
\[ f(x) = a(x + 2)(x + 1)(x - 2) \]

Let's expand this polynomial:

\[ f(x) = a(x + 2)(x + 1)(x - 2) \]
\[ = a(x^2 + x - 2)(x - 2) \]
\[ = a(x^3 - x^2 - 2x^2 + 2x - 2x + 4) \]
\[ = a(x^3 - 3x^2 + 4) \]

Since the graph gives us the point \( B = (0.319, 8.643) \), we can use this to solve for \( a \):

\[ f(0.319) = a(0.319^3 - 3(0.319)^2 + 4) = 8.643 \]

First, calculate \( 0.319^3 \) and \( 3(0.319)^2 \):

\[ 0.319^3 \approx 0.032 \]
\[ 3(0.319)^2 \approx 0.305 \]

Thus,

\[ f(0.319) = a(0.032 - 0.305 + 4) = 8.643 \]
\[ a(3.727) = 8.643 \]
\[ a \approx \frac{8.643}{3.727} \approx 2.32 \]

So, the polynomial \( f(x) \approx 2.32(x^3 - 3x^2 + 4) \).

Now, we evaluate \( f(4) \):

\[ f(4) = 2.32(4^3 - 3(4)^2 + 4) \]
\[ = 2.32(64 - 48 + 4) \]
\[ = 2.32(20) \]
\[ = 46.4 \]

Therefore, the remainder when the polynomial is divided by \( x-4 \) is 46.4.

Would you like more details on any step or have any further questions? Here are some related questions for further exploration:

1. How can you derive the polynomial equation from given points?
2. What is the Remainder Theorem and how is it applied?
3. How do you find the coefficients of a polynomial given specific points?
4. How can you use the roots and their multiplicity to form the polynomial?
5. What is the process of polynomial expansion and simplification?

**Tip:** When evaluating polynomials at specific points, always double-check the calculations for accuracy.

Learn how to find the remainder of a polynomial when divided by x-4 using the Remainder Theorem. This problem involves evaluating a cubic polynomial at a specific point and understanding polynomial division. Suitable for high school students.

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