Question

In Exercises 19–22, match the equivalent expressions. Justify your answers $$ \left(x^2+x+3\right) \div(x-2) $$ A. \(x+1-\frac{1}{x-2}\) B. \(x+3+\frac{9}{x-2}\) C. \(x+1+\frac{5}{x-2}\) D. \(x+3+\frac{3}{x-2}\)

Short answer

The equivalent expression for \((x^2+x+3) \div(x-2)\) is D. \(x+3+\frac{3}{x-2}\)

Step-by-step solution

Divide

To start, we'll perform the division to simplify and rewrite \((x^2+x+3) \div(x-2)\). This can be done using polynomial long division or synthetic division methods. Going ahead with polynomial long division, we get \(x+3+\frac{3}{x-2}\).

Match expressions

Next, we need to match our derived expression with one of the given equivalent expressions. When comparing \(x+3+\frac{3}{x-2}\) with the options A, B, C, and D, we find that it matches with option D.

Justify Answer

Given that our derived expression matches with option D, we can justify it by highlighting that the division process aligns with the polynomial long division process. Therefore, answering option D doesn't contradict with algebra division rules. Thus, the answer is option D.

Key concepts

Equivalent Expressions

Understanding equivalent expressions in algebra is fundamental because they represent the same mathematical quantity, even if they appear different at first glance. When solving polynomial division problems, like the example of \[(x^2+x+3) \div (x-2),\]the goal is to find an expression equivalent to a given one. In this exercise, the division results in several fractional expressions that we need to recognize as equivalent.

Equivalent expressions may look different due to different arrangements or representations. For example:

• \(x+3+\frac{3}{x-2}\) is equivalent to the division of \((x^2+x+3)\) by \((x-2)\)

We can confirm equivalence by substituting common values into each expression, which should yield the same result. Understanding this concept ensures we solve problems accurately and can recognize different forms of the same expression easily.

Synthetic Division

Synthetic division is a quick method used primarily to divide polynomials by linear expressions. It is most efficient when the divisor is in the form \(x-c\). In our exercise involving \[(x^2+x+3) \div (x-2),\]it can be useful for checking the division performed more quickly than polynomial long division.

The process involves:

• Writing down the coefficients of the dividend polynomial.
• Using \(c\) from the divisor \(x-c\), here \(c=2\).
• Performing synthetic substitution to obtain the quotient coefficients.

Using synthetic division here, the operation is simplified and leads to easier identification of quotients, confirming that the expression \[x+3+\frac{3}{x-2}\]is indeed correct.

Polynomial Long Division

Polynomial long division is a detailed process akin to long division with numbers. It is used to divide a high-degree polynomial by another, often when other methods like synthetic division are not applicable. In our example:\[(x^2+x+3) \div (x-2),\]we apply this method to find that the division results in another polynomial plus a remainder.

Steps to perform polynomial long division include:

• Dividing the leading term of the dividend by the leading term of the divisor.
• Multiplying the entire divisor by the result and subtracting from the dividend.
• Repeating the process with the new polynomial till the remainder is of lower degree than the divisor.

Engaging with these steps assures we develop proper solutions and helps verify results from simpler methods like synthetic division, ensuring the accurate result \[x+3+\frac{3}{x-2}.\]

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the basis for composing and simplifying problems within algebra. In our division task:\[(x^2+x+3) \div (x-2),\]the goal was to simplify and rewrite an algebraic expression.

These expressions can be rewritten in various forms without changing their value. In our case:

• We attempted to simplify a quadratic polynomial expression by dividing it by a linear expression.
• The outcome was another simplified algebraic expression \[x+3+\frac{3}{x-2}\]that represents the same initial situation.

By mastering the manipulation of algebraic expressions, students can solve more complex equations and understand the structures within mathematical problems.