5⋅x3+43⋅x2−129⋅x+129=c⋅(x−3)+a⋅(x−3)3+b⋅(x−3)2+d For which a, b, c and d does the following hold for all x ∈ R:



To solve the given polynomial identity:

\[
5x^3 + 43x^2 - 129x + 129 = c(x-3) + a(x-3)^3 + b(x-3)^2 + d
\]

we need to expand the right-hand side and match the coefficients of the corresponding powers of \(x\) with the left-hand side. Let's start by expanding the right-hand side.

### Step 1: Expand \( (x-3)^3 \) and \( (x-3)^2 \)

\[
(x-3)^3 = x^3 - 9x^2 + 27x - 27
\]

\[
(x-3)^2 = x^2 - 6x + 9
\]

### Step 2: Substitute these into the right-hand side expression

Substituting these expansions into the right-hand side, we get:

\[
c(x-3) + a(x-3)^3 + b(x-3)^2 + d = c(x-3) + a(x^3 - 9x^2 + 27x - 27) + b(x^2 - 6x + 9) + d
\]

\[
= ax^3 - 9ax^2 + 27ax - 27a + bx^2 - 6bx + 9b + cx - 3c + d
\]

### Step 3: Combine like terms

Grouping like terms:

\[
= ax^3 + (-9a + b)x^2 + (27a - 6b + c)x + (-27a + 9b - 3c + d)
\]

### Step 4: Compare coefficients with the left-hand side

The left-hand side is:

\[
5x^3 + 43x^2 - 129x + 129
\]

Matching the coefficients of \(x^3\), \(x^2\), \(x\), and the constant term:

1. Coefficient of \(x^3\): \(a = 5\)
2. Coefficient of \(x^2\): \(-9a + b = 43\)
3. Coefficient of \(x\): \(27a - 6b + c = -129\)
4. Constant term: \(-27a + 9b - 3c + d = 129\)

### Step 5: Solve for \( a, b, c, d \)

1. From \(a = 5\):

   Substitute \(a = 5\) into the equations:

2. Solve for \(b\):

   \[
   -9(5) + b = 43 \Rightarrow -45 + b = 43 \Rightarrow b = 88
   \]

3. Solve for \(c\):

   \[
   27(5) - 6(88) + c = -129 \Rightarrow 135 - 528 + c = -129 \Rightarrow c = 264
   \]

4. Solve for \(d\):

   \[
   -27(5) + 9(88) - 3(264) + d = 129
   \]
   \[
   -135 + 792 - 792 + d = 129 \Rightarrow d = 129
   \]

Thus, the values of \(a\), \(b\), \(c\), and \(d\) that satisfy the given polynomial identity are:

\[
\boxed{a = 5, \; b = 88, \; c = 264, \; d = 129}
\]

Would you like more details or have any other questions?

### Related Questions:

1. How can polynomial identities be verified using substitution?
2. What are the steps to expand higher-degree polynomials efficiently?
3. How does one determine if a polynomial is factorable?
4. What is the significance of equating coefficients in polynomial identities?
5. How can polynomial identities be solved using synthetic division?

**Tip:** Always verify your solutions by substituting the values back into the original equation to ensure consistency.

5⋅x^3 + 43⋅x^2 − 129⋅x + 129 = c⋅(x−3) + a⋅(x−3)^3 + b⋅(x−3)^2 + d. For which a, b, c, and d does the following hold for all x ∈ R?

Find the values of a, b, c, and d for which the polynomial identity 5x^3 + 43x^2 - 129x + 129 equals c(x-3) + a(x-3)^3 + b(x-3)^2 + d for all x ∈ R. Detailed step-by-step solution includes expansion, coefficient matching, and solving equations. Suitable for Grades 10-12.

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