The task is to divide the polynomial \( 2x^2 + 7x - 4 \) by \( x - 3 \). This is a classic problem of polynomial long division. Let's solve it step by step, following the "Divide, Multiply, Subtract, Bring Down" method.
Problem:
\[
\frac{2x^2 + 7x - 4}{x - 3}
\]
Step 1: Set up the division
We write the division as:
\[
\begin{array}{r|rrr}
x - 3 & 2x^2 & +7x & -4 \\
\end{array}
\]
Step 2: Divide
- Divide the leading term of the dividend (\(2x^2\)) by the leading term of the divisor (\(x\)):
\[
\frac{2x^2}{x} = 2x
\]
- Write \(2x\) above the division bar.
Step 3: Multiply
- Multiply the entire divisor (\(x - 3\)) by \(2x\):
\[
2x \cdot (x - 3) = 2x^2 - 6x
\]
- Write this result below the dividend:
\[
\begin{array}{r|rrr}
x - 3 & 2x^2 & +7x & -4 \\
& 2x^2 & -6x & \\
\end{array}
\]
Step 4: Subtract
- Subtract the product from the dividend:
\[
(2x^2 + 7x) - (2x^2 - 6x) = 2x^2 + 7x - 2x^2 + 6x = 13x
\]
- Write the result below:
\[
\begin{array}{r|rrr}
x - 3 & 2x^2 & +7x & -4 \\
& 2x^2 & -6x & \\
\hline
& & 13x & -4 \\
\end{array}
\]
Step 5: Bring Down
- Bring down the next term from the dividend, which is \(-4\):
\[
\begin{array}{r|rrr}
x - 3 & 2x^2 & +7x & -4 \\
& 2x^2 & -6x & \\
\hline
& & 13x & -4 \\
\end{array}
\]
Step 6: Repeat the process
- Divide the leading term of the new polynomial (\(13x\)) by the leading term of the divisor (\(x\)):
\[
\frac{13x}{x} = 13
\]
- Write \(13\) above the division bar next to \(2x\).
Step 7: Multiply
- Multiply the entire divisor (\(x - 3\)) by \(13\):
\[
13 \cdot (x - 3) = 13x - 39
\]
- Write this result below:
\[
\begin{array}{r|rrr}
x - 3 & 2x^2 & +7x & -4 \\
& 2x^2 & -6x & \\
\hline
& & 13x & -4 \\
& & 13x & -39 \\
\end{array}
\]
Step 8: Subtract
- Subtract the product from the current polynomial:
\[
(13x - 4) - (13x - 39) = 13x - 4 - 13x + 39 = 35
\]
- Write the result below:
\[
\begin{array}{r|rrr}
x - 3 & 2x^2 & +7x & -4 \\
& 2x^2 & -6x & \\
\hline
& & 13x & -4 \\
& & 13x & -39 \\
\hline
& & & 35 \\
\end{array}
\]
Step 9: Write the final answer
- The quotient is the polynomial above the division bar: \(2x + 13\).
- The remainder is \(35\).
Thus, the result of the division is:
\[
\boxed{2x + 13 + \frac{35}{x - 3}}
\]
This is the final answer in the form of the quotient plus the remainder divided by the divisor.
Parent Tip: Review the logic above to help your child master the concept of dividing polynomials long division worksheet.
