Problem 81 - Elementary Algebra

Question

Problem: Use long division to simplify $$ \left(2 x^{2}+5 x+3\right) \div(x-8) $$ Incorrect Answer: $$ \begin{array}{r} x - 8 \longdiv { 2 x ^ { 2 } + 5 x + 3 } \\ \frac{-\left(x^{2}-8 x\right)}{13 x+3} \\ \frac{-(13 x-104)}{107} \end{array} $$

Step-by-Step Solution

Answer

2x + 21 + \(\frac{171}{x - 8}\)

Step 1 - Setting up the Division

Write the polynomial to be divided, $$2x^2 + 5x + 3$$, under the long division symbol. Place the divisor, $$x - 8$$, outside the long division symbol.

Step 2 - Dividing the First Term

Divide the first term of the dividend, $$2x^2$$, by the first term of the divisor, $$x$$. $$2x^2 ÷ x = 2x$$. Place $$2x$$ in the quotient above the polynomial.

Step 3 - Multiplying

Multiply $$2x$$ by the entire divisor, $$x - 8$$. $$2x \times (x - 8) = 2x^2 - 16x$$. Write $$2x^2 - 16x$$ under the $$2x^2 + 5x + 3$$.

Step 4 - Subtract

Subtract $$2x^2 - 16x$$ from $$2x^2 + 5x + 3$$ to find the remainder. $$2x^2 + 5x + 3 - (2x^2 - 16x) = 21x + 3$$.

Step 5 - Repeat the Process

Divide $$21x$$ by $$x$$ to get $$21$$. Write $$21$$ in the quotient next to $$2x$$. Multiply $$21$$ with the divisor, $$x - 8$$: $$21 \times (x - 8) = 21x - 168$$. Write $$21x - 168$$ under $$21x + 3$$.

Step 6 - Subtraction to Find the Remainder

Subtract $$21x - 168$$ from $$21x + 3$$: $$21x + 3 - (21x - 168) = 171$$. This gives the final remainder.

Step 7 - Write the Final Answer

The final answer is the quotient plus the remainder over the divisor: $$2x + 21 + \frac{171}{x - 8}$$.

Key Concepts

Polynomial Division

Polynomial division is similar to long division with numbers but applied to algebraic expressions.
You start by dividing the highest degree term of the polynomial (dividend) by the highest degree term of the divisor.
In our example, we divided \(2x^2\) by \(x\), resulting in \(2x\).
Next, multiply the entire divisor (\(x - 8\)) by this result, which gives us \(2x^2 - 16x\).
Subtract this from the original polynomial to find the new remainder. Repeat these steps until no further division is possible.
In each step, you simplify until you get a quotient and possibly a remainder.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and performing operations to make the expression easier to understand.
In the polynomial division process, we repeatedly simplify by focusing on the leading terms.
For example, after the first division and multiplication in our exercise, we arrived at \(2x^2 - 16x\).
Subtraction of this from the original dividend gives \(21x + 3\).
This step-by-step breakdown helps to slowly reduce the complexity of the expression.
The result is a more straightforward quotient and a possible remainder that is easier to work with.

Remainder in Algebra

When dividing polynomials, you might end up with a remainder, similar to numerical division.
The remainder is what's left when you can no longer divide the polynomial without getting into fractional terms.
In our problem, after performing all steps of the division, we found a final remainder of 171.
This remainder is then written as a fraction with the original divisor in the denominator:
\(2x + 21 + \frac{171}{x - 8}\).
The remainder indicates that the division isn't exact but provides a detailed understanding of how the original polynomial relates to the divisor.