Problem 76 - Algebra A Combined Function

Question

Multiply or divide as indicated. $$ \left(\frac{x^{2}-9}{x^{2}-1} \cdot \frac{x^{2}+2 x+1}{2 x^{2}+9 x+9}\right) \div \frac{2 x+3}{1-x} $$

Step-by-Step Solution

Answer

The simplified expression is \(-x^2 + 2x + 3\).

Step 1: Rewrite the Division as Multiplication

The given expression involves division by a fraction. To simplify this, you can rewrite the division as multiplication by the reciprocal. \[ \left(\frac{x^{2}-9}{x^{2}-1} \cdot \frac{x^{2}+2x+1}{2x^{2}+9x+9}\right) \div \frac{2x+3}{1-x} \] becomes \[ \left(\frac{x^{2}-9}{x^{2}-1} \cdot \frac{x^{2}+2x+1}{2x^{2}+9x+9}\right) \times \frac{1-x}{2x+3} \]

Step 2: Factor All Polynomials

Factor each polynomial in the numerators and denominators where possible.- \( x^2 - 9 = (x+3)(x-3) \)- \( x^2 - 1 = (x+1)(x-1) \)- \( x^2 + 2x + 1 = (x+1)^2 \)- \( 2x^2 + 9x + 9 \) can be factored as \( (2x+3)(x+3) \)- \( 1-x = -(x-1) \)

Step 3: Substitute Factored Forms into Expression

Substitute the factored forms into the expression:\[ \left(\frac{(x+3)(x-3)}{(x+1)(x-1)} \cdot \frac{(x+1)^2}{(2x+3)(x+3)}\right) \times \frac{-(x-1)}{2x+3} \]

Step 4: Simplify the Expression

Now, cancel common factors from the numerator and the denominator:- \((x+3)\) in the first and second terms cancel out.- \((x-1)\) cancels out between the first and third terms.- \((x+1)\) cancels one of the factors in \((x+1)^2\).- \(2x+3\) in the second and third terms cancel out.The expression simplifies to:\[ -(x-3)(x+1) \]

Step 5: Expand the Result

Multiply the remaining factors:\[ -(x-3)(x+1) = -((x)(x) + (x)(1) - (3)(x) - (3)(1)) \]\[ = -(x^2 + x - 3x - 3) \]Combine and simplify:\[ = -(x^2 - 2x - 3) \]So the final simplified expression is:\[ = -x^2 + 2x + 3 \]

Key Concepts

Polynomial Division

In algebra, polynomial division is a process that helps us simplify expressions where we need to divide one polynomial by another. It is similar to long division with numbers. In polynomial division, we break down higher-degree polynomials into more manageable parts. When faced with a division problem, like dividing fractions, always start by rewriting the division as a multiplication. This involves multiplying by the reciprocal of the divisor.

• To simplify: first change from division to multiplication using the reciprocal.
• This will make the expression easier to work with by turning it into a standard multiplication operation.

By changing the division into multiplication, we can focus on simplifying the expression through factoring, cancellation, and other algebraic processes.

Factoring Polynomials

Factoring polynomials is a key technique in algebra used to simplify polynomials. It involves breaking down a polynomial into a product of simpler polynomials that, when multiplied together, give the original polynomial back. Recognize common types of factoring, such as the difference of squares, which applies to polynomials like \(x^2 - 9\).

• For instance: \(x^2 - 9 = (x+3)(x-3)\) as it's a difference of squares.
• Similarly, \(x^2 - 1 = (x+1)(x-1)\).
• Perfect square trinomials like \(x^2 + 2x + 1\) factor into \((x+1)^2\).

For more complex polynomials, such as \(2x^2 + 9x + 9\), trial and error or techniques like the AC method can be used. Efficient factoring can make the simplification of rational expressions much simpler.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Simplifying these entails finding and cancelling common factors. The process generally includes:

• Factoring polynomials in the numerator and denominator wherever possible.
• Identifying and cancelling out any common factors between them.

Rational expressions can initially seem complex but become easier with practice in factoring. For example, considering the expression \(\frac{x^{2}-9}{x^{2}-1}\), factor each polynomial first before looking for terms that appear in both the numerator and the denominator to simplify.
Simplifying rational expressions is like simplifying fractions in arithmetic but requires extra attention to polynomial factors.

Simplification Techniques

Simplification techniques are crucial in algebra to make expressions manageable and understandable. The goal is to look for common factors across numerators and denominators and cancel them to reduce the expression to its simplest form. Here's a flow:

• Factor all the terms fully.
• Cancel out any common factors between numerators and denominators.
• Look for patterns like difference of squares or trinomial squares.
• Combine remaining elements carefully to avoid mistakes.

For example, in our problem, a sequence of cancelations occurs: - \((x+3)\) cancels between parts of the numerator and denominator.- \((x-1)\) and \((x+1)\) are also canceled to simplify the expression. Once all simplifications are done, what's left should be expanded if necessary, arriving at a neat, concise result. Mastering these techniques enhances your ability to handle even more complex algebraic expressions.