Problem 106 - Elementary Algebra

Question

For the following problems, divide the polynomials. $$ 3 r^{2}-17 r-27 \text { by } r-7 $$

Step-by-Step Solution

Answer

Solution: The quotient is \(r-3\) and the remainder is \(1\).

Step 1: Set up the long division

Write the given polynomial division problem in long division notation, with the dividend (the polynomial being divided) under the long division symbol and the divisor (what you're dividing by) to the left: $$ \begin{array}{c|cc cc} \multicolumn{2}{r}{r} & -3 \\ \cline{2-5} r-7 & 3r^2 & -17r & -27 \\ \cline{2-2} \end{array} $$

Step 2: Divide the first terms

Divide the first term of the dividend (\(3r^2\)) by the first term of the divisor (\(r\)): $$ \frac{3r^2}{r} = 3r $$ Write the result (\(3r\)) in the quotient above the long division symbol.

Step 3: Multiply and subtract

Multiply the divisor (\(r-7\)) by the result obtained in Step 2 (\(3r\)) and subtract this product from the dividend. $$ (3r)(r-7) = 3r^2 - 21r $$ Subtract this product from the dividend: $$ \begin{array}{c|cc cc} \multicolumn{2}{r}{r} & -3 \\ \cline{2-5} r-7 & 3r^2 & -17r & -27 \\ & -(3r^2 & -21r) & \\ \cline{2-3} \multicolumn{2}{r}{\underline{\phantom{xx}}} & 4r & -27 \\ \cline{3-3} \end{array} $$

Step 4: Divide the first remaining terms

Divide the first term of the remaining expression in the dividend (\(4r\)) by the first term of the divisor (\(r\)): $$ \frac{4r}{r} = 4 $$ Write the result (\(4\)) in the quotient above the long division symbol.

Step 5: Multiply and subtract again

Multiply the divisor (\(r-7\)) by the result obtained in Step 4 (\(4\)) and subtract this product from the remaining expression in the dividend: $$ (4)(r-7) = 4r - 28 $$ Subtract this product from the remaining expression: $$ \begin{array}{c|cc cc} \multicolumn{2}{r}{r} & -3 \\ \cline{2-5} r-7 & 3r^2 & -17r & -27 \\ & -(3r^2 & -21r) & \\ \cline{2-3} \multicolumn{2}{r}{\underline{\phantom{xx}}} & 4r & -27 \\ & & -(4r & -28) \\ \cline{3-4} \multicolumn{2}{r}{\underline{\phantom{xx}}} & & & 1 \\ \cline{4-4} \end{array} $$

Step 6: Final answer

The quotient obtained after dividing the polynomial is \(r-3\), and the remainder is \(1\). Therefore, the final answer is: $$ \frac{3r^2 - 17r - 27}{r-7} = r-3 + \frac{1}{r-7} $$

Key Concepts

Long Division

Long division, much like the method used in arithmetic, involves dividing larger expressions into smaller, more manageable parts. You begin by setting up the division symbol, placing the dividend (the polynomial to be divided) underneath and the divisor (the polynomial you are using to divide) outside.

The process involves dividing, multiplying, subtracting, and repeating these steps until you’ve effectively broken down the whole expression. This method is essential for dividing polynomials, as it allows us to find a quotient and remainder just like in basic arithmetic. By practicing long division, students can tackle more complex algebraic equations with ease.

Quotient and Remainder

When dividing polynomials, the result is often expressed in terms of a quotient and a remainder. The quotient is the main result of the division, representing how many times the divisor can be multiplied to stay under the dividend.

The remainder is what is left over after the division process is complete, much like when you learned about division in elementary school. In the provided example, the quotient is "r-3", while the remainder is "1". This remainder is often expressed as a fraction over the original divisor for clarity. Understanding these terms is crucial, as they are the primary results of any polynomial division task.

Divisor and Dividend

The terms divisor and dividend are fundamental in the context of division. In polynomial division, the dividend is the polynomial you want to divide. In our example, it is "3r^2 - 17r - 27".

The divisor is what you divide the dividend by, which in this case is "r-7". Think of the divisor as the element trying to "fit into" the dividend, dividing it into simpler terms. Recognizing which polynomial serves as the divisor and which as the dividend is important, as it sets the stage for the entire division process.

Algebraic Expressions

Algebraic expressions are composed of variables and constants combined using operations like addition, subtraction, multiplication, and division. In polynomial division, these expressions take the form of sums of multiple terms, each with varying degrees of the variable.

For instance, the expression "3r^2 - 17r - 27" is a polynomial, where each term has a different power of the variable "r". Understanding algebraic expressions and how to manipulate them is key in solving polynomial division problems. They allow us to simplify complex problems into manageable steps, using the rules of algebra to guide our process.