Question

polynomial division box method (guided)
score: 0/2 penalty: 1 off
question
fill in the missing values below one at a time to find the quotient when $9x^3 - 6x^2 - 29x - 14$ is divided by $3x + 2$.
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$$\begin{array}{|c|c|c|c|} \\cline{2-4} \\multicolumn{1}{c|}{} & \\square & & \\\\ \\hline 3x & 9x^3 & & \\\\ \\hline +2 & & & \\\\ \\hline \\end{array}$$

answer attempt 1 out of 2
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Explanation:

Step1: Find the first term of the quotient

To find the first term of the quotient, we divide the leading term of the dividend \(9x^3\) by the leading term of the divisor \(3x\). So, \(\frac{9x^3}{3x} = 3x^2\). This is the first term in the top - left box of the box method (the first term of the quotient).

Step2: Multiply the divisor by the first term of the quotient

Now, we multiply the divisor \(3x + 2\) by \(3x^2\).

• Multiply \(3x\) by \(3x^2\): \(3x\times3x^2=9x^3\) (which is the first term in the second row, first column, and this matches the given value).
• Multiply \(2\) by \(3x^2\): \(2\times3x^2 = 6x^2\). This will be the first term in the third row, first column.

Step3: Subtract the product from the dividend (for the next term)

The next term of the dividend is \(-6x^2\). After subtracting the product of the divisor and the first term of the quotient (the \(6x^2\) we just found) from \(-6x^2\), we get \(-6x^2-6x^2=-12x^2\). Now, we divide \(-12x^2\) by \(3x\) (the leading term of the divisor) to get the second term of the quotient: \(\frac{-12x^2}{3x}=-4x\). This is the second term in the top row.

Step4: Multiply the divisor by the second term of the quotient

Multiply \(3x + 2\) by \(-4x\).

• Multiply \(3x\) by \(-4x\): \(3x\times(-4x)=-12x^2\) (this goes in the second row, second column).
• Multiply \(2\) by \(-4x\): \(2\times(-4x)=-8x\) (this goes in the third row, second column).

Step5: Subtract the product from the remaining dividend

The next term of the dividend is \(-29x\). After subtracting the product of the divisor and the second term of the quotient (the \(-8x\) we just found) from \(-29x\), we get \(-29x-(-8x)=-29x + 8x=-21x\). Now, we divide \(-21x\) by \(3x\) (the leading term of the divisor) to get the third term of the quotient: \(\frac{-21x}{3x}=-7\). This is the third term in the top row.

Step6: Multiply the divisor by the third term of the quotient

Multiply \(3x + 2\) by \(-7\).

• Multiply \(3x\) by \(-7\): \(3x\times(-7)=-21x\) (this goes in the second row, third column).
• Multiply \(2\) by \(-7\): \(2\times(-7)=-14\) (this goes in the third row, third column). And when we subtract, we see that there is no remainder.

The quotient is \(3x^2-4x - 7\). The first missing value (the first term of the quotient) is \(3x^2\).

Answer:

The first missing value (the value in the top - left box) is \(3x^2\)