1.1 Q-5

Question Statement

Given the function $f(x) = x^3 - ax^2 + bx + 1$, where $f(2) = -3$ and $f(-1) = 0$, find the values of $a$ and $b$.

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Background and Explanation

To solve this problem, we will use the fact that the function value is given for specific inputs, $x = 2$ and $x = -1$. By substituting these values into the function, we create a system of equations that will help us solve for the unknowns $a$ and $b$.

We also need to recall how to evaluate a polynomial at a specific point and how to solve a system of linear equations.

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Solution

We are given the function: $f(x) = x^3 - ax^2 + bx + 1$

Step 1: Substitute $x = 2$ into the equation.

Using the given $f(2) = -3$, substitute $x = 2$ into the function:

$$f(2) = 2^3 - a(2^2) + b(2) + 1 = -3$$

Simplify:

$$8 - 4a + 2b + 1 = -3$$ $$9 - 4a + 2b = -3$$

Rearrange this equation:

$$-4a + 2b = -12 \tag{1}$$

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Step 2: Substitute $x = -1$ into the equation.

Now, using $f(-1) = 0$, substitute $x = -1$ into the function:

$$f(-1) = (-1)^3 - a(-1)^2 + b(-1) + 1 = 0$$

Simplify:

$$-1 - a - b + 1 = 0$$ $$-a - b = 0 \quad \Rightarrow \quad a + b = 0 \tag{2}$$

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Step 3: Solve the system of equations.

We now have the system of equations:

1. $-4a + 2b = -12$
2. $a + b = 0$

From equation (2), solve for $b$:

$$b = -a$$

Substitute $b = -a$ into equation (1):

$$-4a + 2(-a) = -12$$

Simplify:

$$-4a - 2a = -12$$ $$-6a = -12$$

Solve for $a$:

$$a = 2$$

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Step 4: Find the value of $b$.

Now that we know $a = 2$, substitute it into $a + b = 0$:

$$2 + b = 0$$ $$b = -2$$

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Key Formulas or Methods Used

• Substitution to evaluate polynomials at specific points.
• Solving systems of linear equations.

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Summary of Steps

1. Substitute $x = 2$ into the function to get an equation involving $a$ and $b$.
2. Substitute $x = -1$ into the function to get another equation.
3. Solve the system of equations:
   • From $a + b = 0$, find $b = -a$.
   • Substitute $b = -a$ into the first equation and solve for $a$.
4. Substitute the value of $a$ into $a + b = 0$ to find $b$.
5. The solution is $a = 2$ and $b = -2$.