Question

question
fill in the missing values below one at a
time to find the quotient when
$4x^3 + 11x^2 + 7x + 2$ is divided by
$x + 2$.
\

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Explanation:

Step1: Find the first term of the quotient

To divide \(4x^3 + 11x^2 + 7x + 2\) by \(x + 2\), we start with the leading term. Divide \(4x^3\) by \(x\) to get \(4x^2\). So the first term of the quotient (top row, first box) is \(4x^2\).

Step2: Multiply \(x + 2\) by \(4x^2\)

Multiply \(x\) by \(4x^2\) to get \(4x^3\) (already given in the table), and multiply \(2\) by \(4x^2\) to get \(8x^2\). So the box in the \("+2"\) row and first column (under \(4x^3\)) is \(8x^2\).

Step3: Subtract and find the next term

Subtract \(8x^2\) from \(11x^2\) to get \(3x^2\). Now, divide \(3x^2\) by \(x\) to get \(3x\). This is the second term of the quotient (top row, second box: \(3x\)).

Step4: Multiply \(x + 2\) by \(3x\)

Multiply \(x\) by \(3x\) to get \(3x^2\) (this goes in the \(x\) row, second box), and multiply \(2\) by \(3x\) to get \(6x\) (this goes in the \("+2"\) row, second box).

Step5: Subtract and find the next term

Subtract \(6x\) from \(7x\) to get \(x\). Divide \(x\) by \(x\) to get \(1\). This is the third term of the quotient (top row, third box: \(1\)).

Step6: Multiply \(x + 2\) by \(1\)

Multiply \(x\) by \(1\) to get \(x\) (this goes in the \(x\) row, third box), and multiply \(2\) by \(1\) to get \(2\) (this goes in the \("+2"\) row, third box).

Step7: Verify the remainder

Subtract \(2\) from \(2\) to get \(0\), so there's no remainder.

Now, filling in the table:

• Top row (quotient): \(4x^2\), \(3x\), \(1\)
• \(x\) row (after first term): \(3x^2\), \(x\) (wait, let's re - check the multiplication steps properly for the \(x\) row boxes:
• After the first term \(4x^2\), when we have the second term \(3x\) in the quotient, multiplying \(x\) by \(3x\) gives \(3x^2\) (second box in \(x\) row). Then with the third term \(1\) in the quotient, multiplying \(x\) by \(1\) gives \(x\) (third box in \(x\) row).
• \(+2\) row:
• First box: \(8x^2\) (from \(2\times4x^2\))
• Second box: \(6x\) (from \(2\times3x\))
• Third box: \(2\) (from \(2\times1\))

Let's present the filled - in table values step - by - step as per the calculation:

Top row (quotient):

First box: \(4x^2\) (from dividing \(4x^3\) by \(x\))
Second box: \(3x\) (from dividing \(3x^2\) (after subtraction \(11x^2 - 8x^2\)) by \(x\))
Third box: \(1\) (from dividing \(x\) (after subtraction \(7x - 6x\)) by \(x\))

\(x\) row (multiplication by \(x\)):

First box: \(4x^3\) (given)
Second box: \(3x^2\) (from \(x\times3x\))
Third box: \(x\) (from \(x\times1\))

\(+2\) row (multiplication by \(2\)):

First box: \(8x^2\) (from \(2\times4x^2\))
Second box: \(6x\) (from \(2\times3x\))
Third box: \(2\) (from \(2\times1\))

If we are just asked for the quotient (the top row values combined), the quotient is \(4x^2+3x + 1\). But if we are filling the table:

Top row (left to right): \(4x^2\), \(3x\), \(1\)

\(x\) row (middle column, then right column): \(3x^2\), \(x\)

\(+2\) row (left to right): \(8x^2\), \(6x\), \(2\)

Answer:

The quotient is \(4x^2 + 3x+1\) (and the filled - in table values are as calculated above). If we consider the quotient polynomial, the answer is \(4x^{2}+3x + 1\)