Problem 43 - Beginning and Intermediate Algebra

Question

Divide. $$\left(2 x^{3}-7 x^{2}-2 x+10\right) \div(2 x+5)$$

Step-by-Step Solution

Answer

The result of dividing \(\left(2 x^{3}-7 x^{2}-2 x+10\right) \div(2 x+5)\) using long division is: \[x^{2}-3x+1 - \frac{5}{2x+5}\]

Step 1: Setting up the Long Division

First, we'll set up the long division of the polynomial expression: $$ \begin{array}{c|cc cc} \multicolumn{2}{r}{x^2} & -3x & +1 \\ \cline{2-5} 2x+5 & 2x^3 & -7x^2 & -2x & +10\\ \cline{2-5} \end{array} $$

Step 2: Divide the First Terms

Divide the first term of the dividend (inside) by the first term of the divisor (outside), \(2x^3\) divided by \(2x\), and write the result above the dividend: $$ \begin{array}{c|cc cc} x^2 & \\ \cline{2-5} 2x+5 & 2x^3 & -7x^2 & -2x & +10\\ \cline{2-5} \end{array} $$

Step 3: Multiplying and Subtracting

Now, multiply the divisor by the term we just found (\(x^2\)), and then subtract the result from the dividend $$ \begin{array}{c|cc cc} x^2 & \\ \cline{2-5} 2x+5 & 2x^3 & -7x^2 & -2x & +10\\ && (-)2x^3 & -5x^2 \\ \cline{2-5} &&&& \end{array} $$

Step 4: Bring Down the Next Term

Next, bring down the next term of the dividend (-2x) and write it to the right of the remainder $$ \begin{array}{c|cc cc} x^2 & \\ \cline{2-5} 2x+5 & 2x^3 & -7x^2 & -2x & +10\\ && (-)2x^3 & -5x^2 \\ \cline{2-5} && & -2x & \\ \end{array} $$

Step 5: Divide the New Dividend

Divide the new dividend, -2x, by the first term of the divisor, 2x $$ \begin{array}{c|cc cc} x^2 & -3 & \\ \cline{2-5} 2x+5 & 2x^3 & -7x^2 & -2x & +10\\ && (-)2x^3 & -5x^2 \\ \cline{2-5} && & -2x & \\ \end{array} $$

Step 6: Multiply, Subtract and Bring down

Multiply the divisor by -3, and subtract the result from the new dividend. Then bring down the next term in the dividend (+10) $$ \begin{array}{c|cc cc} x^2 & -3 & \\ \cline{2-5} 2x+5 & 2x^3 & -7x^2 & -2x & +10\\ && (-)2x^3 & -5x^2 \\ \cline{2-5} && & -2x & \\ &&& (+)2x & +15 \\ \cline{4-5} &&&& -5 \end{array} $$

Step 7: Express The Final Result

The division process is now complete. Write down the quotient with the remainder, if any. The result of the division process is: \(x^2 - 3x + 1\) with the remainder of \(-5\). So, the final answer is: \[ \left(2 x^{3}-7 x^{2}-2 x+10\right) \div(2 x+5) = x^{2}-3x+1 - \frac{5}{2x+5} \]

Key Concepts

Remainder Theorem

The remainder theorem gives us a simple way to find the remainder of a polynomial when it is divided by a linear divisor of the form \(x-c\). According to the theorem:

• If \(f(x)\) is a polynomial of degree \(n\) and it is divided by \(x-c\), then the remainder of this division is \(f(c)\).

Let's apply this to a specific example. Consider the polynomial \(f(x) = 2x^3 - 7x^2 - 2x + 10\) divided by, say \(x+1\). The remainder can be found by simply evaluating \(f(-1)\), since the divisor is \(x - (-1)\).

The remainder theorem is especially useful for quickly checking the results of polynomial division, offering a fast method to verify the work without having to perform lengthy calculations.

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear polynomial of the form \(x-c\). It's usually faster and requires less writing than traditional long division, making it an appealing method especially for simpler calculations.

• First, write down the coefficients of the dividend polynomial.
• Then, use the zero of the divisor (the value \(c\) from \(x-c\)), which is negative, in synthetic division.
• Combine these values through multiplication and addition.

Returning to our original division problem, \(2x^3 - 7x^2 - 2x + 10\) divided by \(2x + 5\), synthetic division can simplify steps into quick calculations by focusing just on the coefficients and constant term part, making sure steps are followed in sequence correctly to avoid errors. This method is especially useful for students who are comfortable with basic arithmetic and seek a faster method of performing polynomial division.

Polynomial Division Steps

When dividing a polynomial by another polynomial, like in our example, you can follow these steps:

• Setup: Write the dividend (the polynomial you're dividing) and the divisor in a long division setup.
• Divide the Leading Terms: Take the first term of the dividend and divide it by the first term of the divisor. Write that result as part of the quotient.
• Multiply and Subtract: Multiply the entire divisor by this newly found part of the quotient, then subtract from the original dividend to get a new dividend.
• Bring Down the Next Term: Move down the next coefficient's term from the original polynomial. Repeat from the Divide step.

Continue the repeat process of dividing, multiplying, and subtracting until every term in the original polynomial has been used. The result will include the quotient and possibly a remainder, as seen in the example \(x^2 - 3x + 1\) with a remainder of \(-5\). This method provides a systematic approach to polynomial division, ensuring accuracy and completeness in finding the solution.
Understanding each step helps demystify the division process, so you can approach similar problems with confidence.