Problem 29 - Algebra 1: Concepts and Skills

Question

FACTORING AFTER ADDING OR SUBTRACTING. Simplify the expression. $$ \frac{2 x(x+4)}{(x+1)^{2}}-\frac{3 x-3}{(x+1)^{2}} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \[ \frac{2x + 3}{x + 1}\]

Step 1: Combine the fractions

Since the two fractions have the same denominator which is \((x+1)^{2}\), we can subtract the second numerator from the first one. This gives us \[\frac{2x(x + 4) - (3x -3)}{(x+1)^{2}}\]

Step 2: Distribute the numbers to the variables

Distribute the 2x into 'x' and '4' in the first term, we'll also distribute '-1' to '3x' and '-3' in the second term which results in: \[\frac{2x^{2}+8x-3x+3}{(x+1)^{2}}\]

Step 3: Simplify the numerator

Combine the like terms in the numerator, as a result we get: \[\frac{2x^{2}+5x+3}{(x+1)^{2}}\]

Step 4: Factor the numerator

The expression in the numerator is a quadratic trinomial, which can be factored. We end up with \[\frac{(2x+3)(x+1)}{(x+1)^{2}}\]

Step 5: Cancel out common factors

(x+1) is a common factor in the numerator and the denominator, so it can be removed. Note that the denominator will still have (x+1) left: \[ \frac{2x + 3}{x + 1}\]

Key Concepts

Factoring

Factoring involves breaking down an expression into simpler parts called factors, which multiply together to give the original expression. When presented with an algebraic expression, note that factoring can greatly simplify the process of solving or simplifying it.

• Consider a polynomial like the one in our problem: it often contains terms that can be divided out by finding a common factor.
• For example, if twice the numerator of a fraction equals some polynomial, find potential factors that multiply to restore the number.

Factoring can also involve pulling out common elements or finding what two numbers multiply together to produce a constant term, all done with the aim to simplify and solve the expression.

Combining Like Terms

When we talk about combining like terms, we’re referring to the process of simplifying an expression by adding or subtracting terms that have identical variable parts.

• Look for terms that have the same variables raised to the same power.
• For instance, in the expression \(2x^2 + 8x - 3x + 3\), '8x' and '-3x' are like terms.
• Add or subtract the coefficients of these like terms while keeping the variables unchanged.

Combining like terms is a fundamental step to reduce the complexity of polynomials, which is very useful when simplifying mathematical expressions.

Distributive Property

The Distributive Property in mathematics allows you to remove parentheses by distributing a factor across terms inside the parentheses. This property is quite useful when you want to simplify expressions that initially appear complicated.

• For example, in \(2x(x + 4)\), you multiply '2x' by both 'x' and '4' separately.
• This gives us the expression \(2x^2 + 8x\).
• Similarly, distributing '-1' across \(3x - 3\) yields \(-3x + 3\).

Using the distributive property effectively lays the foundation for further simplifications, enabling you to work with expressions more easily.

Quadratic Trinomial

A quadratic trinomial is a polynomial with three terms where the highest degree is two. These are generally represented in the form \(ax^2 + bx + c\).

• In the context of the original problem, the expression \(2x^2 + 5x + 3\) is a quadratic trinomial.
• To factor it, look for two numbers that multiply to \(ac\) (here 2*3 = 6) and add up to \(b\) (which is 5).
• These numbers help in breaking down the middle term and factoring the trinomial into two binomial expressions.

Factoring quadratic trinomials is a skillful activity that helps in revealing the roots of the expression, and is a predictive method to find when the expression should equal zero.

Canceling Common Factors

Once you've factored expressions, identifying and canceling common factors is the next logical step. This involves reducing the complexity of an expression by eliminating terms that appear in both the numerator and the denominator.

• Consider, \((x+1)\) was both in the numerator and denominator in our exercise.
• We could safely 'cancel' it, which simplified the expression to \(\frac{2x + 3}{x + 1}\).
• This technique greatly reduces algebraic fractions to their simplest form.

Understanding how to identify and cancel common factors is essential in making algebraic expressions much simpler and easier to solve.