X^4-x^3+ax^2+x+b, is divisible by x+2,x-3 find a,b say if a+2b=5

We are given that the polynomial \( P(x) = x^4 - x^3 + ax^2 + x + b \) is divisible by \( (x + 2) \) and \( (x - 3) \). This means that both \( x = -2 \) and \( x = 3 \) are roots of the polynomial, i.e.,

\[
P(-2) = 0 \quad \text{and} \quad P(3) = 0.
\]

### Step 1: Substitute \( x = -2 \) into the polynomial

\[
P(-2) = (-2)^4 - (-2)^3 + a(-2)^2 + (-2) + b = 0
\]
\[
16 + 8 + 4a - 2 + b = 0
\]
\[
22 + 4a + b = 0
\]
\[
4a + b = -22 \quad \text{(Equation 1)}
\]

### Step 2: Substitute \( x = 3 \) into the polynomial

\[
P(3) = 3^4 - 3^3 + a(3^2) + 3 + b = 0
\]
\[
81 - 27 + 9a + 3 + b = 0
\]
\[
57 + 9a + b = 0
\]
\[
9a + b = -57 \quad \text{(Equation 2)}
\]

### Step 3: Solve the system of equations

From Equation 1:
\[
b = -22 - 4a
\]

Substitute this into Equation 2:
\[
9a + (-22 - 4a) = -57
\]
\[
9a - 22 - 4a = -57
\]
\[
5a = -35
\]
\[
a = -7
\]

Substitute \( a = -7 \) back into Equation 1:
\[
4(-7) + b = -22
\]
\[
-28 + b = -22
\]
\[
b = 6
\]

### Step 4: Check if \( a + 2b = 5 \)

Substitute \( a = -7 \) and \( b = 6 \):
\[
a + 2b = -7 + 2(6) = -7 + 12 = 5
\]
Thus, \( a + 2b = 5 \) is true.

### Final Answer:
\[
a = -7, \quad b = 6
\]
And \( a + 2b = 5 \) is true.

Would you like any more details or have further questions?

Here are 5 related questions:
1. How do you verify if a polynomial is divisible by a binomial?
2. What does it mean for a number to be a root of a polynomial?
3. How do you solve a system of equations using substitution?
4. Can a polynomial have multiple roots?
5. What is the Remainder Theorem and how is it applied?

**Tip**: Always check your solutions by substituting values back into the original equations to avoid mistakes.

The query submitted by the user: X^4 - x^3 + ax^2 + x + b is divisible by x+2 and x-3. Find a and b, and verify if a + 2b = 5.

Learn how to find the values of a and b such that the polynomial X^4 - x^3 + ax^2 + x + b is divisible by x+2 and x-3. This step-by-step solution covers polynomial division, the Remainder Theorem, and solving a system of equations. Suitable for students in Grades 10-12.

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