Problem 33 - Algebra for College Students

Question

Perform the indicated divisions. $$ \frac{4 a^{2}-8 a b+4 b^{2}}{a-b} $$

Step-by-Step Solution

Answer

The simplified expression is \(4a - 4b\).

Step 1: Recognize the Format

Notice that the expression in the numerator, \(4a^2 - 8ab + 4b^2\), resembles a quadratic trinomial. Look for patterns or familiar factorizations.

Step 2: Factor the Numerator

Recognize that the numerator is a perfect square trinomial. It can be factored as \((2a - 2b)^2\). This is because \((2a - 2b)(2a - 2b)\) results in \(4a^2 - 8ab + 4b^2\).

Step 3: Simplify the Fraction

With the numerator factored as \((2a - 2b)(2a - 2b)\), the division becomes \(\frac{(2a - 2b)(2a - 2b)}{a-b}\). Notice that \((2a - 2b)\) can be written as \(2(a-b)\). So, the fraction simplifies to \(\frac{(2(a-b))(2(a-b))}{a-b}\).

Step 4: Cancel Common Factors

Cancel \((a-b)\) from both the numerator and the denominator. This leaves us with \(2(2(a-b))\) after cancellation, which simplifies to \(2(2a - 2b)\).

Step 5: Final Simplification

Simplify \(2(2a - 2b)\) to \(4a - 4b\). This is the simplified expression after performing the division.

Key Concepts

Quadratic Trinomial

A quadratic trinomial is a polynomial expression with three terms where the highest degree is two. This means you will have terms in the format of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) isn’t zero. In this particular exercise, the quadratic trinomial presented is \(4a^2 - 8ab + 4b^2\). Understanding the structure of a quadratic trinomial will help you identify patterns and potential factorizations.

When you see something like this, it often indicates a perfect square trinomial—a trinomial that can be factored into a product of binomials that are the same. Recognizing perfect square trinomials assists in simplified polynomial division by reducing complex expressions into manageable parts.

• Look for common factors in the terms.
• Check symmetry in the coefficients of the first and last term.
• Consider completing the square if necessary.

Factoring Polynomials

Factoring polynomials involves rewriting them as a product of simpler polynomials, making equations easier to work with. In this case, you need to factor \(4a^2 - 8ab + 4b^2\) into binomials. This trinomial is actually a perfect square that can be expressed as \((2a - 2b)^2\).

Here’s a simple way to factor:

• Identify a common pattern in the polynomial.
• Use patterns such as \(a^2 - 2ab + b^2 = (a-b)^2\) for perfect squares.
• Check by multiplying the factors back to ensure that you receive the original polynomial.

This step simplifies the division process, making cancellation of common factors possible. Factoring is a critical algebraic tool that makes polynomial operations easier and more straightforward.

Algebraic Simplification

Algebraic Simplification is about reducing complex expressions into their simplest forms. In this exercise, the division simplifies significantly once you factor the quadratic trinomial. First, factor the numerator: \((2a-2b)^2\) which can be depicted as \(2(a-b) \times 2(a-b)\).

Next, set up the division: \((2(a-b) \times 2(a-b))/(a-b)\). Here, you can cancel one \(a-b\) from the numerator and the denominator because it's common in both.

• Initially write out the expression.
• Identify any common factors across the numerator and the denominator.
• Cancel the common factors.
• Simplify what's left to achieve a concise expression.

After cancellation, the expression is reduced to \(2(2(a-b))\) which simplifies further to \((4a - 4b)\). This process highlights the power of simplification in problem-solving, delivering clear and clean results.