-x^3+8x^2-9x+2=(Ax+B)(x-3)^2+C(x-3)(x^2+1)+D(x^2+1) find A,B,C,D

We are tasked with finding the values of \( A \), \( B \), \( C \), and \( D \) such that:

\[
-x^3 + 8x^2 - 9x + 2 = (Ax + B)(x - 3)^2 + C(x - 3)(x^2 + 1) + D(x^2 + 1)
\]

### Step 1: Expand both sides

#### Left-hand side: 
\[
-x^3 + 8x^2 - 9x + 2
\]

#### Right-hand side:
We will expand the right-hand side term by term.

##### 1. Expand \((Ax + B)(x - 3)^2\):
\[
(x - 3)^2 = x^2 - 6x + 9
\]
Now, multiply \((Ax + B)\) by this quadratic:
\[
(Ax + B)(x^2 - 6x + 9) = Ax^3 - 6Ax^2 + 9Ax + Bx^2 - 6Bx + 9B
\]
Simplifying this gives:
\[
Ax^3 + (-6A + B)x^2 + (9A - 6B)x + 9B
\]

##### 2. Expand \(C(x - 3)(x^2 + 1)\):
First, distribute \( (x - 3) \):
\[
C(x - 3)(x^2 + 1) = C(x^3 + x - 3x^2 - 3)
\]
Simplifying this gives:
\[
C(x^3 - 3x^2 + x - 3)
\]

##### 3. Expand \(D(x^2 + 1)\):
\[
D(x^2 + 1) = Dx^2 + D
\]

### Step 2: Combine all terms on the right-hand side

Now, sum up all the expanded terms:
\[
Ax^3 + (-6A + B)x^2 + (9A - 6B)x + 9B
+ Cx^3 - 3Cx^2 + Cx - 3C
+ Dx^2 + D
\]
Grouping like terms:
\[
(A + C)x^3 + (-6A + B - 3C + D)x^2 + (9A - 6B + C)x + (9B - 3C + D)
\]

### Step 3: Equate coefficients with the left-hand side

Now, compare the coefficients of \( x^3 \), \( x^2 \), \( x \), and the constant term with the left-hand side \( -x^3 + 8x^2 - 9x + 2 \).

- Coefficient of \( x^3 \): \( A + C = -1 \)
- Coefficient of \( x^2 \): \( -6A + B - 3C + D = 8 \)
- Coefficient of \( x \): \( 9A - 6B + C = -9 \)
- Constant term: \( 9B - 3C + D = 2 \)

### Step 4: Solve the system of equations

We now solve the following system of equations:

1. \( A + C = -1 \)
2. \( -6A + B - 3C + D = 8 \)
3. \( 9A - 6B + C = -9 \)
4. \( 9B - 3C + D = 2 \)

Let's solve this step by step.

The solution to the system of equations is:

\[
A = -1, \quad B = 0, \quad C = 0, \quad D = 2
\]

Thus, the original equation becomes:
\[
-x^3 + 8x^2 - 9x + 2 = (-x)(x - 3)^2 + 2(x^2 + 1)
\]

Would you like further clarification on any part of this solution, or a more detailed breakdown?

Here are 5 related questions to consider:
1. What steps are necessary to expand a cubic expression like \( (Ax + B)(x - 3)^2 \)?
2. How does comparing coefficients help in solving for unknowns in an equation?
3. Can this method be applied to solve higher-degree polynomials?
4. What if one of the terms on the right-hand side involved an additional power of \( x \)?
5. How does solving systems of equations help in other areas of algebra?

**Tip:** Always group like terms carefully to avoid errors when expanding and comparing expressions in polynomial equations.

-x^3 + 8x^2 - 9x + 2 = (Ax + B)(x - 3)^2 + C(x - 3)(x^2 + 1) + D(x^2 + 1), find A, B, C, D

Explore how to solve the equation -x^3 + 8x^2 - 9x + 2 = (Ax + B)(x - 3)^2 + C(x - 3)(x^2 + 1) + D(x^2 + 1). This detailed solution involves polynomial expansion, comparing coefficients, and solving systems of equations. Perfect for high school students in Grades 10-12.

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