Problem 40 - Beginning and Intermediate Algebra

Question

Divide. $$\left(50 c^{3}+7 c+4\right) \div(5 c+2)$$

Step-by-Step Solution

Answer

The short answer to the division \[\left(50 c^{3}+7 c+4\right) \div(5 c+2)\] is \(10c^2 - 4c + \frac{15c + 4}{5c + 2}\).

Step 1: Set up the long division

Write the dividend (the polynomial being divided) and the divisor (the polynomial dividing) in the long division format: \[ \require{enclose} \begin{array}{c|cc cc c} \multicolumn{2}{r}{\underline{\phantom{xx}}} & & \\ \cline{2-6} 5 c + 2 & 50c^3 & + & 7 c & + & 4 \\ \end{array} \]

Step 2: Divide the first term of the dividend by the first term of the divisor

Divide \(50c^3\) by \(5c\), which results in \(10c^2\). Write this term above the division: \[ \require{enclose} \begin{array}{c|cc cc c} \multicolumn{2}{r}{10c^2 } & & \\ \cline{2-6} 5 c + 2 & 50c^3 & + & 7 c & + & 4 \\ \end{array} \]

Step 3: Multiply the divisor by the quotient term

Multiply the divisor, \(5 c + 2\), by the quotient term, \(10c^2\). This results in \(50c^3 + 20c^2\). Write this result below the dividend: \[ \require{enclose} \begin{array}{c|cc cc c} \multicolumn{2}{r}{10c^2 } & & \\ \cline{2-6} 5 c + 2 & 50c^3 & + & 7 c & + & 4 \\ &-(50c^3 & + & 20c^2) & & \\ \cline{2-4} \end{array} \]

Step 4: Subtract

Subtract the product obtained in step 3 from the dividend: \[ \require{enclose} \begin{array}{c|c cc cc c} \multicolumn{2}{r}{10c^2 } & & \\ \cline{2-6} 5 c + 2 & 50c^3 & + & 7 c & + & 4 \\ &-(50c^3 & + & 20c^2) & & \\ \cline{2-4} & & & -20c^2 &+ &7c \\ \end{array} \]

Step 5: Bring down the next term

Bring down the next term from the dividend, which is \(+4\): \[ \require{enclose} \begin{array}{c|c cc cc c} \multicolumn{2}{r}{10c^2 } & & \\ \cline{2-6} 5 c + 2 & 50c^3 & + & 7 c & + & 4 \\ &-(50c^3 & + & 20c^2) & & \\ \cline{2-4} & & & -20c^2 &+ &7c \\ &&&&+&4 \\ \end{array} \]

Step 6: Divide the first term of the new dividend by the first term of the divisor

Divide \(-20c^2\) by \(5c\), which results in \(-4c\). Write this term above the division: \[ \require{enclose} \begin{array}{c|c cc cc c} \multicolumn{2}{r}{10c^2 } &- & 4c \\ \cline{2-6} 5 c + 2 & 50c^3 & + & 7 c & + & 4 \\ &-(50c^3 & + & 20c^2) & & \\ \cline{2-4} & & & -20c^2 &+ &7c \\ &&&&+&4 \\ \end{array} \]

Step 7: Multiply the divisor by the new quotient term

Multiply the divisor, \(5 c + 2\), by the new quotient term, \(-4c\). This results in \(-20c^2 - 8c\). Write this result below the new dividend: \[ \require{enclose} \begin{array}{c|c cc cc c} \multicolumn{2}{r}{10c^2 } &- & 4c \\ \cline{2-6} 5 c + 2 & 50c^3 & + & 7 c & + & 4 \\ &-(50c^3 & + & 20c^2) & & \\ \cline{2-4} & & & -20c^2 &+ &7c \\ & & & +(-20c^2 &- & 8c) \\ \cline{4-5} \end{array} \]

Step 8: Subtract

Subtract the product obtained in step 7 from the new dividend: \[ \require{enclose} \begin{array}{c|c cc cc c} \multicolumn{2}{r}{10c^2 } &- & 4c \\ \cline{2-6} 5 c + 2 & 50c^3 & + & 7 c & + & 4 \\ &-(50c^3 & + & 20c^2) & & \\ \cline{2-4} & & & -20c^2 &+ &7c \\ & & & +(-20c^2 &- & 8c) \\ \cline{4-5} & & & & &15c &+ &4 \\ \end{array} \] Since the degree of the remaining polynomial (15c + 4) is lower than the degree of the divisor (5c + 2), we can stop the division process. The remaining polynomial is the remainder. The final quotient is \(10c^2 - 4c\), and the remainder is \(15c + 4\). Therefore, the division expression can be written as \[(50 c^{3}+7 c+4) \div(5 c+2) = 10c^2 - 4c + \frac{15c + 4}{5c + 2}.\]

Key Concepts

Dividend and Divisor

In polynomial long division, the terms **dividend** and **divisor** are foundational. The **dividend** is the polynomial that you aim to divide. It's the larger expression at the beginning of your problem. Meanwhile, the **divisor** is the polynomial by which you divide the dividend. For example, in the problem \((50c^3 + 7c + 4) \div (5c + 2)\), the dividend is \(50c^3 + 7c + 4\), and the divisor is \(5c + 2\). Remember:

• **Dividend**: what's being divided.
• **Divisor**: what's doing the dividing.

Setting up these terms correctly is crucial as it determines how you'll proceed through the division steps. Always align terms by their degree, which helps in maintaining order during the process.

Division of Polynomials

The process of dividing polynomials is analogous to long division with numbers. It involves several steps of dividing, multiplying, subtracting, and bringing down terms. To divide polynomials, you start by matching the highest degree term of your dividend with the highest degree term of your divisor. This step helps in determining your initial term in the quotient.

1. **Divide** the leading term of the dividend by the leading term of the divisor to find the first term of the quotient. In our example, divide \(50c^3\) by \(5c\) to get \(10c^2\).
2. **Multiply** this quotient term by the entire divisor and write the result beneath the dividend.
3. **Subtract** this result from the dividend, similar to numerical long division.
4. **Bring down** the next term from the original dividend and repeat these steps until every term has been accounted for or cannot be divided any further.
Through iteration, you systematically complete the expression. This sequence of steps enables a clear transition in solving the polynomial division problem.

Remainder in Polynomial Division

After completing the division process, if the polynomial you're left with (which is lower in degree than the divisor) cannot be further divided by the divisor, it's called the remainder. Think of it like having left-over pieces that can't fit into equal groups anymore. In our given exercise, the remainder is what is left after those repeated steps of dividing, multiplying, and subtracting are done.

• The **remainder** here is \(15c + 4\), since it is not possible to further divide it by the divisor \(5c + 2\).

Whenever you write down your final answer, it will be in the form: \[dividend = (divisor \times quotient) + remainder.\]This expression captures not just the quotient but also the remainder, giving a comprehensive answer to the original division problem. Understanding the remainder's role highlights that not all polynomials divide evenly, which is an important aspect of polynomial division.