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To solve the equation [tex]\((p x + q)\left(x^2 - 3 x + 4\right) = A x^3 + B x^2 + C x + D\)[/tex] and find the values of [tex]\(A\)[/tex], [tex]\(B\)[/tex], [tex]\(C\)[/tex], and [tex]\(D\)[/tex] given that [tex]\(B = C\)[/tex], follow these steps:
### Step 1: Expanding the Expression
Expand the expression [tex]\((p x + q)\left(x^2 - 3 x + 4\right)\)[/tex] to find the coefficients of the resulting polynomial.
[tex]\[ (p x + q)\left(x^2 - 3 x + 4\right) = p x \left(x^2 - 3 x + 4\right) + q \left(x^2 - 3 x + 4\right) \][/tex]
[tex]\[ = p x^3 - 3 p x^2 + 4 p x + q x^2 - 3 q x + 4 q \][/tex]
### Step 2: Combining Like Terms
Combine the terms with the same degree of [tex]\(x\)[/tex]:
[tex]\[ p x^3 + (-3 p + q) x^2 + (4 p - 3 q) x + 4 q \][/tex]
### Step 3: Identifying Coefficients
From the expanded expression, we can identify the coefficients:
- The coefficient of [tex]\(x^3\)[/tex] is [tex]\(p\)[/tex].
- The coefficient of [tex]\(x^2\)[/tex] is [tex]\(-3 p + q\)[/tex].
- The coefficient of [tex]\(x\)[/tex] is [tex]\(4 p - 3 q\)[/tex].
- The constant term is [tex]\(4 q\)[/tex].
Therefore, we have:
[tex]\[ A = p \][/tex]
[tex]\[ B = -3 p + q \][/tex]
[tex]\[ C = 4 p - 3 q \][/tex]
[tex]\[ D = 4 q \][/tex]
### Step 4: Equating [tex]\(B\)[/tex] and [tex]\(C\)[/tex]
Given that [tex]\(B = C\)[/tex], we set the coefficients of [tex]\(x^2\)[/tex] and [tex]\(x\)[/tex] equal to each other:
[tex]\[ -3 p + q = 4 p - 3 q \][/tex]
### Step 5: Solving for [tex]\(p\)[/tex] and [tex]\(q\)[/tex]
Rearrange the equation to solve for [tex]\(p\)[/tex] in terms of [tex]\(q\)[/tex]:
[tex]\[ -3 p + q = 4 p - 3 q \][/tex]
[tex]\[ q + 3 q = 4 p + 3 p \][/tex]
[tex]\[ 4 q = 7 p \][/tex]
[tex]\[ p = \frac{4 q}{7} \][/tex]
So, the values of [tex]\(p\)[/tex] and [tex]\(q\)[/tex] that satisfy the condition [tex]\(B = C\)[/tex] are:
[tex]\[ p = \frac{4 q}{7} \][/tex]
The solution in terms of [tex]\(p\)[/tex] and [tex]\(q\)[/tex] is:
[tex]\[ (p, q) = \left(\frac{4 q}{7}, q\right) \][/tex]
Thus, for the given question where [tex]\(B\)[/tex] equals [tex]\(C\)[/tex], the value of [tex]\(p\)[/tex] and [tex]\(q\)[/tex] is [tex]\(\left(\frac{4 q}{7}, q\right)\)[/tex].
### Step 1: Expanding the Expression
Expand the expression [tex]\((p x + q)\left(x^2 - 3 x + 4\right)\)[/tex] to find the coefficients of the resulting polynomial.
[tex]\[ (p x + q)\left(x^2 - 3 x + 4\right) = p x \left(x^2 - 3 x + 4\right) + q \left(x^2 - 3 x + 4\right) \][/tex]
[tex]\[ = p x^3 - 3 p x^2 + 4 p x + q x^2 - 3 q x + 4 q \][/tex]
### Step 2: Combining Like Terms
Combine the terms with the same degree of [tex]\(x\)[/tex]:
[tex]\[ p x^3 + (-3 p + q) x^2 + (4 p - 3 q) x + 4 q \][/tex]
### Step 3: Identifying Coefficients
From the expanded expression, we can identify the coefficients:
- The coefficient of [tex]\(x^3\)[/tex] is [tex]\(p\)[/tex].
- The coefficient of [tex]\(x^2\)[/tex] is [tex]\(-3 p + q\)[/tex].
- The coefficient of [tex]\(x\)[/tex] is [tex]\(4 p - 3 q\)[/tex].
- The constant term is [tex]\(4 q\)[/tex].
Therefore, we have:
[tex]\[ A = p \][/tex]
[tex]\[ B = -3 p + q \][/tex]
[tex]\[ C = 4 p - 3 q \][/tex]
[tex]\[ D = 4 q \][/tex]
### Step 4: Equating [tex]\(B\)[/tex] and [tex]\(C\)[/tex]
Given that [tex]\(B = C\)[/tex], we set the coefficients of [tex]\(x^2\)[/tex] and [tex]\(x\)[/tex] equal to each other:
[tex]\[ -3 p + q = 4 p - 3 q \][/tex]
### Step 5: Solving for [tex]\(p\)[/tex] and [tex]\(q\)[/tex]
Rearrange the equation to solve for [tex]\(p\)[/tex] in terms of [tex]\(q\)[/tex]:
[tex]\[ -3 p + q = 4 p - 3 q \][/tex]
[tex]\[ q + 3 q = 4 p + 3 p \][/tex]
[tex]\[ 4 q = 7 p \][/tex]
[tex]\[ p = \frac{4 q}{7} \][/tex]
So, the values of [tex]\(p\)[/tex] and [tex]\(q\)[/tex] that satisfy the condition [tex]\(B = C\)[/tex] are:
[tex]\[ p = \frac{4 q}{7} \][/tex]
The solution in terms of [tex]\(p\)[/tex] and [tex]\(q\)[/tex] is:
[tex]\[ (p, q) = \left(\frac{4 q}{7}, q\right) \][/tex]
Thus, for the given question where [tex]\(B\)[/tex] equals [tex]\(C\)[/tex], the value of [tex]\(p\)[/tex] and [tex]\(q\)[/tex] is [tex]\(\left(\frac{4 q}{7}, q\right)\)[/tex].
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