Question

When dividing polynomials if the remainder is zero (no remainder) then the divisor is a factor. Use synthetic division to determine if the first expression is a factor of the second expression:First expression: x+1 Second expression: x^3+2x+3 If it is a factor then just type the quotient as the answer. If it is not a factor then just type "no". Be sure to type your answer in descending powers of x with now spaces between your terms. Use the "^" key (shift+6) to indicate a power/exponent.Answer:

Answer 1

$\begin{gathered} \text{Given} \\ \frac{x^3+2x+3}{x+1} \end{gathered}$$\begin{gathered} \text{Step 1 Divide the leading term of the dividend by the leading term of the divisor:} \\ \dfrac{x^{3}}{x}=x^2 \\ \text{Write down the calculated result in the upper part of the table.} \\ \text{Multiply it by the divisor: }x^2(x+1)=x^3+x^2 \\ \text{Subtract the dividend from the obtained result:} \\ (x^3+2x+3)-(x^3+x^2)=-x^2+2x+3 \end{gathered}$$\begin{gathered} \text{Step 2 Divide the leading term of the dividend by the leading term of the divisor:} \\ \dfrac{- x^{2}}{x}=-x \\ \text{Write down the calculated result in the upper part of the table.} \\ \text{Multiply it by the divisor: }-x(x+1)=-x^2-x \\ \text{Subtract the dividend from the obtained result:} \\ (x^3+2x+3)-(-x^2-x)=3x+3 \end{gathered}$$\begin{gathered} \text{Step 3 Divide the leading term of the dividend by the leading term of the divisor:} \\ \dfrac{3 x}{x}=3 \\ \text{Write down the calculated result in the upper part of the table.} \\ \text{Multiply it by the divisor: }3(x+1)=3x+3 \\ \text{Subtract the dividend from the obtained result:} \\ (x^3+2x+3)-(3x+3)=0 \end{gathered}$

This results in a division table of

$\begin{gathered} \text{Therefore, the quotient is }x^2-x+3 \\ \end{gathered}$