To find the quotient of \( -12x^3 - 23x^2 + 15x - 24 \) divided by \( -3x - 8 \) using the box method (also known as the area model), we can start by setting up the box and finding the missing values step by step.
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Set Up the Box: The divisor is \( -3x - 8 \) and the dividend is \( -12x^3 - 23x^2 + 15x - 24 \). The first box will correspond to the leading term of the divisor and the leading term of the dividend.
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Finding the First Term of the Quotient: We need to find what multiplies \( -3x \) to give \( -12x^3 \). We can do this using polynomial long division or by inspection: \[ -3x \cdot (4x^2) = -12x^3 \] Therefore, the first term in the quotient is \( 4x^2 \).
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Multiplying and Subtracting: Next, we multiply \( 4x^2 \) by the entire divisor: \[ 4x^2 \cdot (-3x) = -12x^3 \] \[ 4x^2 \cdot (-8) = -32x^2 \] Now, we subtract these results from the original polynomial: \[ (-12x^3 - 23x^2 + 15x - 24) - (-12x^3 - 32x^2) = (32 - 23)x^2 + 15x - 24 = 9x^2 + 15x - 24 \]
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Finding the Next Term of the Quotient: Next, we find what multiplies \( -3x \) to give \( 9x^2 \): \[ -3x \cdot (-3) = 9x^2 \] Thus, the next term in the quotient is \( -3 \).
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Final Multiplication and Subtraction: We multiply: \[ -3 \cdot (-3x) = 9x \] \[ -3 \cdot (-8) = 24 \] We subtract these from what we have: \[ (9x^2 + 15x - 24) - (9x + 24) = 15x - 9x - 24 - 24 = 6x - 48 \]
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Final Box Filling: From this process, we find that:
- The first term of the quotient is \( 4x^2 \) (to fill the first box).
- The second term of the quotient is \( -3 \) (to fill the second box). The remainder we got is \( 6x - 48 \), which can be further simplified if necessary.
Thus, the quotient when \( -12x^3 - 23x^2 + 15x - 24 \) is divided by \( -3x - 8 \) is: \[ \boxed{4x^2 - 3} \]
After filling in both boxes and subtracting correctly, you can verify the full polynomial long division process to ensure accuracy. The final area representing the answer should reflect this quotient along with any remainder if applicable.