Problem 42 - Algebra 2

Question

Simplify each expression. \(\frac{a^{2}+2 a+1}{2 a^{2}+3 a+1}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{a+1}{2a+1}\).

Step 1: Understand the Rational Expression

The given expression is a rational expression, which means it is a fraction where both the numerator and the denominator are polynomials. Our task is to simplify it, if possible.

Step 2: Factor the Numerator

The numerator is given by the expression \(a^2 + 2a + 1\). This is a perfect square trinomial and can be factored as \((a+1)^2\). This simplification is based on the identity \((x+y)^2 = x^2 + 2xy + y^2\).

Step 3: Factor the Denominator

The denominator is the polynomial \(2a^2 + 3a + 1\). We need to factor this expression as well. By inspection or using the quadratic formula, we factor it into \((2a+1)(a+1)\).

Step 4: Simplify the Expression

Now we have the fraction \(\frac{(a+1)^2}{(2a+1)(a+1)}\). Since \((a+1)\) is common in both the numerator and denominator, we can cancel it out, resulting in the expression \(\frac{a+1}{2a+1}\).

Step 5: Verify Simplification

The expression \(\frac{a+1}{2a+1}\) is the simplest form, assuming \(a eq -1\) (to avoid division by zero). Always verify such conditions when canceling terms.

Key Concepts

Factoring Polynomials

Factoring polynomials is like breaking down a large, complex puzzle into smaller, easier pieces. When you have a polynomial, your goal is to express it as a product of its factors. This involves identifying numbers or expressions that can be multiplied together to get back the original polynomial.

• Look for common factors first. See if there's a number or variable that is common in all terms of the polynomial.


• Identify special patterns, such as difference of squares or perfect square trinomials, which have known factoring formulas.


• For quadratic polynomials (like those with degree 2), find factors of the constant term that sum up to the middle term coefficient.


• Use the quadratic formula if factoring by simple methods seems difficult.

Practicing these steps can help simplify complex polynomials into their factored forms, aiding in solving and simplifying rational expressions.

Perfect Square Trinomials

Perfect square trinomials are a unique type of polynomial that can be expressed as the square of a binomial. Recognizing these trinomials helps in quickly factoring and simplifying expressions.
A perfect square trinomial follows the pattern

• First term is square: The first term should be something squared, for example, in \(a^2 + 2a + 1\), \(a^2\) is a perfect square.


• Middle term: The middle term should be twice the product of the terms inside the binomial. In \\(a^2 + 2a + 1\), 2a is \(2 \times a \times 1\).


• Last term is square:The last term should also be a perfect square. In \(a^2 + 2a + 1\), 1 is \(1^2\).

This trinomial, \(a^2 + 2a + 1\), can be rewritten as \((a+1)^2\), which simplifies the factorization process significantly.

Quadratic Formula

The quadratic formula is a crucial tool for solving quadratic equations that may not be easily factorable by inspection. It allows you to find the roots of the equation \[ax^2 + bx + c = 0.\] The formula is:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]

• The quadratic formula can be used to factor polynomials when other factoring methods fail.


• "Discriminant" tells whether the roots are real or complex, found inside the square root: \\(b^2 - 4ac\).


• If the discriminant is positive, you'll have two real roots. If it's zero, one real root. And if negative, the roots are complex.


• Applying the quadratic formula to \\(2a^2 + 3a + 1\) provided us the factors \ (2a+1)(a+1).

Thus, understanding the quadratic formula is essential for solving and simplifying quadratic polynomials.

Fraction Simplification

Fraction simplification is all about reducing fractions to their simplest form. A rational expression, like any fraction, is simplified by dividing out common factors from the numerator and the denominator. Here's how you can do it:

• **Identify common factors:** Look for terms that are identical in both the numerator and the denominator.


• **Cancel out common terms:** Removing these identical factors simplifies the fraction. For example, \(\frac{(a+1)^2}{(2a+1)(a+1)}\) as \\(a+1\) is common, we can reduce it to
  \(\frac{a+1}{2a+1}\).


• **Check for restrictions:** Ensure the simplified expression doesn't violate any original conditions, such as division by zero (e.g., \ a eq -1 \in this case).

Simplification helps students better understand and work with rational expressions by focusing on the core components of the problem.