12.4 - Division of Algebraic Expressions Continued

In this section, we continue our discussion on the division of algebraic expressions. Division of algebraic expressions involves simplifying expressions by dividing one polynomial by another. The process can be done through methods like long division and synthetic division.

When dividing algebraic expressions, the key is to ensure that each term in the dividend is divided by the divisor, and like terms are combined wherever possible. Let's break down the process with examples.

1. Long Division Method

The long division method is similar to the division of numbers. It involves dividing the first term of the dividend by the first term of the divisor, multiplying, and subtracting the result from the dividend. The process is repeated for the remaining terms until no more terms are left to divide.

For example, dividing the polynomial $x^3 + 2x^2 + 3x + 4$ by $x + 1$:

Step 1: Divide the first term: $x^3 \div x = x^2$.
Step 2: Multiply: $x^2(x + 1) = x^3 + x^2$.
Step 3: Subtract: $(x^3 + 2x^2 + 3x + 4) - (x^3 + x^2) = x^2 + 3x + 4$.
Step 4: Divide the first term: $x^2 \div x = x$.
Step 5: Multiply: $x(x + 1) = x^2 + x$.
Step 6: Subtract: $(x^2 + 3x + 4) - (x^2 + x) = 2x + 4$.
Step 7: Divide the first term: $2x \div x = 2$.
Step 8: Multiply: $2(x + 1) = 2x + 2$.
Step 9: Subtract: $(2x + 4) - (2x + 2) = 2$.

The quotient is $x^2 + x + 2$ and the remainder is $2$. Therefore, the division can be written as:

2. Synthetic Division

Synthetic division is a faster and more compact method of dividing polynomials, but it only works when dividing by a linear binomial of the form $x - a$. In this method, we use the coefficients of the dividend polynomial and perform a series of multiplications and additions to find the quotient and remainder.

For example, dividing $x^3 + 2x^2 + 3x + 4$ by $x - 1$:

Step 1: Write down the coefficients of the dividend: $[1, 2, 3, 4]$.
Step 2: Use the root of the divisor ($x - 1$ gives root $1$) and set up synthetic division.
Step 3: Bring down the first coefficient (1).
Step 4: Multiply the root (1) by the value just brought down (1), giving 1. Add this to the next coefficient: $2 + 1 = 3$.
Step 5: Multiply 1 by 3, giving 3. Add to the next coefficient: $3 + 3 = 6$.
Step 6: Multiply 1 by 6, giving 6. Add to the next coefficient: $4 + 6 = 10$.
Step 7: The final result is the quotient $x^2 + 3x + 6$ with remainder 10.

Thus, the division result is:

3. Division Involving Terms with Variables

When dividing expressions involving variables, the same process applies, but we need to consider the exponents of the variables. For example, dividing $x^4 + 2x^3 + x^2$ by $x^2$:

Step 1: Divide the first term: $x^4 \div x^2 = x^2$.
Step 2: Multiply: $x^2(x^2) = x^4$.
Step 3: Subtract: $(x^4 + 2x^3 + x^2) - x^4 = 2x^3 + x^2$.
Step 4: Divide the first term: $2x^3 \div x^2 = 2x$.
Step 5: Multiply: $2x(x^2) = 2x^3$.
Step 6: Subtract: $(2x^3 + x^2) - 2x^3 = x^2$.
Step 7: Divide the first term: $x^2 \div x^2 = 1$.
Step 8: Multiply: $1(x^2) = x^2$.
Step 9: Subtract: $(x^2) - x^2 = 0$.

The quotient is $x^2 + 2x + 1$ and there is no remainder. Therefore, the result is:

4. Key Points to Remember

• Always divide each term of the dividend by the divisor.
• In long division, carefully subtract the product from the dividend after each multiplication step.
• Synthetic division is a shortcut method for dividing by a linear factor of the form $x - a$.
• In case of division involving variables, ensure that the exponents of the terms are properly handled.