https://file.aiquickdraw.com/mathbot/file/1728332330029dwzsucwt.jpg

The equation in the image shows:

\[
(x - 4) \cdot P(x) = x^3 - 2x^2 + ax + b
\]

We are asked to determine the value of \( b \).

To solve this, let's express \( P(x) \) as a polynomial and use polynomial division or equating coefficients. Let \( P(x) \) be a quadratic polynomial:

\[
P(x) = x^2 + px + q
\]

Now, expand \( (x - 4) \cdot P(x) \):

\[
(x - 4)(x^2 + px + q) = x^3 + px^2 + qx - 4x^2 - 4px - 4q
\]

Simplify the expression:

\[
x^3 + (p - 4)x^2 + (q - 4p)x - 4q
\]

Now, compare this with the original equation:

\[
x^3 - 2x^2 + ax + b
\]

By equating the coefficients of the corresponding powers of \( x \), we get the following system of equations:

1. Coefficient of \( x^2 \): \( p - 4 = -2 \) → \( p = 2 \)
2. Coefficient of \( x^1 \): \( q - 4p = a \) → \( q - 8 = a \)
3. Constant term: \( -4q = b \)

Now, we know that \( p = 2 \), so substitute into the third equation for \( b \):

\[
-4q = b
\]

Thus, \( b = -4q \).

If you would like further clarification or details on the next steps, feel free to ask!

Here are some related questions to consider:
1. What happens if we try a different degree for \( P(x) \)?
2. How does the value of \( a \) affect the function in this equation?
3. Could there be multiple values for \( b \) in similar polynomial equations?
4. How can polynomial long division help in these types of problems?
5. What if the given equation had no constant term—how would that affect \( P(x) \)?

**Tip:** Always align the degrees of terms when expanding polynomial expressions for easier comparison.

(x - 4)P(x) = x^3 - 2x^2 + ax + b. Find the value of b.

Solve the polynomial equation (x - 4)P(x) = x^3 - 2x^2 + ax + b to find the constant b by expanding and comparing coefficients. This problem is suitable for Grades 10-12, covering polynomial expansion and coefficient comparison.

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