Hasil bagi

p open parentheses x close parentheses equals open parentheses negative 4 times a plus b close parentheses x cubed plus open parentheses 7 times a plus b close parentheses x squared plus 2 times x minus 87

oleh open parentheses x minus 2 close parentheses adalah q open parentheses x close parentheses dengan sisa negative 7. Jika q open parentheses x close parentheses dibagi oleh open parentheses x minus 3 close parentheses bersisa 189, maka nilai 8 times a plus b equals....

Given:
\[
p(x) = (-4a + b)x^3 + (7a + b)x^2 + 2x - 87
\]

### Step-by-Step Breakdown

1. **Dividing \(p(x)\) by \((x - 2)\):**

   According to the problem statement, when \(p(x)\) is divided by \((x - 2)\), the remainder is \(-7\). By the Remainder Theorem:
   \[
   p(2) = -7
   \]

   Substitute \(x = 2\) into \(p(x)\):
   \[
   p(2) = (-4a + b)(2)^3 + (7a + b)(2)^2 + 2(2) - 87
   \]
   Simplifying each term:
   \[
   p(2) = (-4a + b)(8) + (7a + b)(4) + 4 - 87
   \]
   \[
   p(2) = -32a + 8b + 28a + 4b + 4 - 87
   \]
   Combine like terms:
   \[
   p(2) = (-32a + 28a) + (8b + 4b) + 4 - 87
   \]
   \[
   p(2) = -4a + 12b - 83
   \]
   Set this equal to the remainder:
   \[
   -4a + 12b - 83 = -7
   \]
   Solve for \(a\) and \(b\):
   \[
   -4a + 12b = 76 \quad \text{(Equation 1)}
   \]

2. **Dividing \(q(x)\) by \((x - 3)\):**

   When \(q(x)\) is divided by \((x - 3)\), the remainder is 189. Since \(q(x)\) is the quotient of \(p(x)\) divided by \((x - 2)\), and by the polynomial relationship:
   \[
   p(x) = (x - 2)q(x) - 7
   \]
   Then, applying the Remainder Theorem for \(q(x)\) when divided by \((x - 3)\):
   \[
   q(3) = 189
   \]

   Since \(p(x) = (x - 2)q(x) - 7\), substitute \(x = 3\) into this equation:
   \[
   p(3) = (3 - 2)q(3) - 7
   \]
   \[
   p(3) = 1 \times 189 - 7
   \]
   \[
   p(3) = 182
   \]

3. **Calculate \(p(3)\):**

   Substitute \(x = 3\) into the polynomial \(p(x)\):
   \[
   p(3) = (-4a + b)(3)^3 + (7a + b)(3)^2 + 2(3) - 87
   \]
   Simplifying each term:
   \[
   p(3) = (-4a + b)(27) + (7a + b)(9) + 6 - 87
   \]
   \[
   p(3) = -108a + 27b + 63a + 9b + 6 - 87
   \]
   Combine like terms:
   \[
   p(3) = (-108a + 63a) + (27b + 9b) + 6 - 87
   \]
   \[
   p(3) = -45a + 36b - 81
   \]
   Set this equal to the value found earlier:
   \[
   -45a + 36b - 81 = 182
   \]
   Solve for \(a\) and \(b\):
   \[
   -45a + 36b = 263 \quad \text{(Equation 2)}
   \]

4. **Solve the system of equations:**

   From Equation 1:
   \[
   -4a + 12b = 76
   \]
   Divide by 4:
   \[
   -a + 3b = 19
   \]
   Rearrange:
   \[
   a = 3b - 19
   \]

   Substitute into Equation 2:
   \[
   -45(3b - 19) + 36b = 263
   \]
   Expand:
   \[
   -135b + 855 + 36b = 263
   \]
   Combine like terms:
   \[
   -99b + 855 = 263
   \]
   Subtract 855 from both sides:
   \[
   -99b = -592
   \]
   Divide by \(-99\):
   \[
   b = \frac{592}{99} = \frac{8}{9}
   \]

   Substitute \(b = \frac{8}{9}\) into the expression for \(a\):
   \[
   a = 3 \times \frac{8}{9} - 19
   \]
   \[
   a = \frac{24}{9} - 19
   \]
   \[
   a = \frac{24}{9} - \frac{171}{9} = -\frac{147}{9} = -16.33\]

. Therefore

Hasil bagi p(x) = (-4a + b)x^3 + (7a + b)x^2 + 2x - 87 oleh (x - 2) adalah q(x) dengan sisa -7. Jika q(x) dibagi oleh (x - 3) bersisa 189, maka nilai 8a + b = ....

Learn how to solve a polynomial division problem where p(x) = (-4a + b)x^3 + (7a + b)x^2 + 2x - 87 is divided by (x - 2) and (x - 3), with step-by-step explanations using the Remainder Theorem and system of equations. This detailed solution is perfect for high school students in Grades 10-12.

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