Questions: Fill in the missing values below one at a time to find the quotient when (8 x^3+22 x^2+11 x+15) is divided by (2 x+5). (8 x^3) (2 x) (8 x^3) +5

Transcript text: Fill in the missing values below one at a time to find the quotient when $8 x^{3}+22 x^{2}+11 x+15$ is divided by $2 x+5$. \begin{tabular}{|c|c|c|c|} \cline { 2 - 4 } & $\square$ & $\square$ & $\square$ \\ \hline $2 x$ & $8 x^{3}$ & $\square$ & $\square$ \\ \hline+5 & $\square$ & $\square$ & $\square$ \\ \hline \end{tabular}

Solution

Solution Steps

To solve the problem of dividing the polynomial $8x^3 + 22x^2 + 11x + 15$ by $2x + 5$ using the box method, we will follow these steps:

Set up the division by writing the dividend and divisor.
Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.
Multiply the entire divisor by this term and subtract from the dividend.
Repeat the process with the new polynomial formed after subtraction until the degree of the remainder is less than the degree of the divisor.
The terms obtained in each step form the quotient.

Step 1: Define the Polynomials

We start with the dividend $8x^3 + 22x^2 + 11x + 15$ and the divisor $2x + 5$.

Step 2: Perform Polynomial Division

To find the quotient, we divide the leading term of the dividend $8x^3$ by the leading term of the divisor $2x$: \[ \frac{8x^3}{2x} = 4x^2 \] This gives us the first term of the quotient.

Step 3: Multiply and Subtract

Next, we multiply the entire divisor $2x + 5$ by $4x^2$: \[ 4x^2(2x + 5) = 8x^3 + 20x^2 \] Now, we subtract this from the original dividend: \[ (8x^3 + 22x^2 + 11x + 15) - (8x^3 + 20x^2) = 2x^2 + 11x + 15 \]

Step 4: Repeat the Process

Now we repeat the process with the new polynomial $2x^2 + 11x + 15$. We divide the leading term $2x^2$ by $2x$: \[ \frac{2x^2}{2x} = x \] This gives us the second term of the quotient.

Next, we multiply the divisor by $x$: \[ x(2x + 5) = 2x^2 + 5x \] Subtracting this from $2x^2 + 11x + 15$: \[ (2x^2 + 11x + 15) - (2x^2 + 5x) = 6x + 15 \]

Step 5: Final Division

We now divide the leading term $6x$ by $2x$: \[ \frac{6x}{2x} = 3 \] This gives us the final term of the quotient.

We multiply the divisor by $3$: \[ 3(2x + 5) = 6x + 15 \] Subtracting this from $6x + 15$: \[ (6x + 15) - (6x + 15) = 0 \] Thus, there is no remainder.