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To find the points of interaction (denoted as [tex]\(\delta\)[/tex]) between the functions [tex]\(f(x) = (x-1)(x-2)(x+1)\)[/tex] and [tex]\(g(x) = x^3-4\)[/tex], we need to determine where these two functions are equal to each other. In other words, we need to solve the equation:
[tex]\[ f(x) = g(x) \][/tex]
Substituting the given functions, this becomes:
[tex]\[ (x-1)(x-2)(x+1) = x^3 - 4 \][/tex]
Our goal is to find the values of [tex]\(x\)[/tex] that satisfy this equation. Here is the detailed step-by-step solution:
### Step 1: Set up the equation
We need to find the roots of the equation:
[tex]\[ (x-1)(x-2)(x+1) - (x^3 - 4) = 0 \][/tex]
### Step 2: Expand [tex]\(f(x)\)[/tex]
First, expand the expression for [tex]\(f(x)\)[/tex]:
[tex]\[ (x-1)(x-2)(x+1) \][/tex]
First, multiply the first two terms:
[tex]\[ (x-1)(x-2) = x^2 - 3x + 2 \][/tex]
Now, multiply this result by [tex]\((x+1)\)[/tex]:
[tex]\[ (x^2 - 3x + 2)(x+1) = x^3 + x^2 - 3x^2 - 3x + 2x + 2 = x^3 - 2x^2 - x + 2 \][/tex]
So, we have:
[tex]\[ f(x) = x^3 - 2x^2 - x + 2 \][/tex]
### Step 3: Set up the interaction equation
Now we set up the equation [tex]\(f(x) = g(x)\)[/tex]:
[tex]\[ x^3 - 2x^2 - x + 2 = x^3 - 4 \][/tex]
Subtract [tex]\(x^3\)[/tex] from both sides of the equation:
[tex]\[ -2x^2 - x + 2 = -4 \][/tex]
### Step 4: Simplify the equation
Simplify by adding 4 to both sides of the equation:
[tex]\[ -2x^2 - x + 6 = 0 \][/tex]
This can be rewritten as:
[tex]\[ 2x^2 + x - 6 = 0 \][/tex]
### Step 5: Solve the quadratic equation
Solve the quadratic equation [tex]\(2x^2 + x - 6 = 0\)[/tex] using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 2\)[/tex], [tex]\(b = 1\)[/tex], and [tex]\(c = -6\)[/tex]:
[tex]\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 2 \cdot (-6)}}{2 \cdot 2} \][/tex]
[tex]\[ x = \frac{-1 \pm \sqrt{1 + 48}}{4} \][/tex]
[tex]\[ x = \frac{-1 \pm \sqrt{49}}{4} \][/tex]
[tex]\[ x = \frac{-1 \pm 7}{4} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{-1 + 7}{4} = \frac{6}{4} = \frac{3}{2} \][/tex]
and
[tex]\[ x = \frac{-1 - 7}{4} = \frac{-8}{4} = -2 \][/tex]
### Step 6: Points of Interaction
Thus, the points [tex]\((\delta)\)[/tex] of interaction between the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] are:
[tex]\[ x = -2 \quad \text{and} \quad x = \frac{3}{2} \][/tex]
So, the values for the points of interaction are:
[tex]\[ \boxed{-2 \quad \text{and} \quad \frac{3}{2}} \][/tex]
[tex]\[ f(x) = g(x) \][/tex]
Substituting the given functions, this becomes:
[tex]\[ (x-1)(x-2)(x+1) = x^3 - 4 \][/tex]
Our goal is to find the values of [tex]\(x\)[/tex] that satisfy this equation. Here is the detailed step-by-step solution:
### Step 1: Set up the equation
We need to find the roots of the equation:
[tex]\[ (x-1)(x-2)(x+1) - (x^3 - 4) = 0 \][/tex]
### Step 2: Expand [tex]\(f(x)\)[/tex]
First, expand the expression for [tex]\(f(x)\)[/tex]:
[tex]\[ (x-1)(x-2)(x+1) \][/tex]
First, multiply the first two terms:
[tex]\[ (x-1)(x-2) = x^2 - 3x + 2 \][/tex]
Now, multiply this result by [tex]\((x+1)\)[/tex]:
[tex]\[ (x^2 - 3x + 2)(x+1) = x^3 + x^2 - 3x^2 - 3x + 2x + 2 = x^3 - 2x^2 - x + 2 \][/tex]
So, we have:
[tex]\[ f(x) = x^3 - 2x^2 - x + 2 \][/tex]
### Step 3: Set up the interaction equation
Now we set up the equation [tex]\(f(x) = g(x)\)[/tex]:
[tex]\[ x^3 - 2x^2 - x + 2 = x^3 - 4 \][/tex]
Subtract [tex]\(x^3\)[/tex] from both sides of the equation:
[tex]\[ -2x^2 - x + 2 = -4 \][/tex]
### Step 4: Simplify the equation
Simplify by adding 4 to both sides of the equation:
[tex]\[ -2x^2 - x + 6 = 0 \][/tex]
This can be rewritten as:
[tex]\[ 2x^2 + x - 6 = 0 \][/tex]
### Step 5: Solve the quadratic equation
Solve the quadratic equation [tex]\(2x^2 + x - 6 = 0\)[/tex] using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 2\)[/tex], [tex]\(b = 1\)[/tex], and [tex]\(c = -6\)[/tex]:
[tex]\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 2 \cdot (-6)}}{2 \cdot 2} \][/tex]
[tex]\[ x = \frac{-1 \pm \sqrt{1 + 48}}{4} \][/tex]
[tex]\[ x = \frac{-1 \pm \sqrt{49}}{4} \][/tex]
[tex]\[ x = \frac{-1 \pm 7}{4} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{-1 + 7}{4} = \frac{6}{4} = \frac{3}{2} \][/tex]
and
[tex]\[ x = \frac{-1 - 7}{4} = \frac{-8}{4} = -2 \][/tex]
### Step 6: Points of Interaction
Thus, the points [tex]\((\delta)\)[/tex] of interaction between the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] are:
[tex]\[ x = -2 \quad \text{and} \quad x = \frac{3}{2} \][/tex]
So, the values for the points of interaction are:
[tex]\[ \boxed{-2 \quad \text{and} \quad \frac{3}{2}} \][/tex]
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