Problem 30 - Algebra A Combined Function

Question

Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. Then factor the trinomial. $$ x^{2}-8 x $$

Step-by-Step Solution

Answer

Add 16 to the binomial; factor to \((x - 4)^2\).

Step 1: Identify the Coefficient

For the binomial given, \(x^2 - 8x\), identify the coefficient of the \(x\) term, which is \(-8\).

Step 2: Divide the Coefficient by 2

Divide the coefficient of \(x\) by 2 to prepare for completing the square: \[ \frac{-8}{2} = -4 \].

Step 3: Square the Result

Square the result from Step 2 to find the constant to add to the binomial: \((-4)^2 = 16\).

Step 4: Form the Perfect Square Trinomial

Add the squared result obtained in Step 3 to the binomial: \(x^2 - 8x + 16\). This forms a perfect square trinomial.

Step 5: Factor the Trinomial

Factor the trinomial \(x^2 - 8x + 16\) into \((x - 4)^2\).

Step 6: Verify the Factorization

Expand \((x - 4)^2\) to ensure it matches the original trinomial. Calculate: \((x - 4)^2 = (x - 4)(x - 4) = x^2 - 8x + 16\), confirming it is correct.

Key Concepts

Factoring

Factoring is an essential skill in algebra that helps break down expressions into simpler components, known as factors. When you factor an expression, such as a perfect square trinomial, you find numbers or expressions that multiply together to give the original expression. This is like unraveling a puzzle to find what makes up the expression.

Let's take the perfect square trinomial \(x^2 - 8x + 16\) as an example. By factoring, we want to rewrite it as the square of a binomial. The factored form of this trinomial is \((x - 4)^2\). This means that when multiplied out, \((x - 4) \cdot (x - 4)\) will give us the original trinomial.

Factoring is not just about numbers but also involves understanding patterns and using algebraic identities like \((a-b)^2 = a^2 - 2ab + b^2\), which is helpful in identifying perfect square trinomials.

Completing the Square

Completing the square is a technique used to make a quadratic expression into a perfect square trinomial. This technique is particularly useful because it helps in solving quadratic equations and in converting them into vertex form.

To complete the square for the expression \(x^2 - 8x\), you should follow these steps:

• Identify the coefficient of the linear term, which is \-8.
• Divide this coefficient by 2: \(-8 \div 2 = -4\).
• Square the result: \((-4)^2 = 16\).

By adding 16 to \(x^2 - 8x\), the expression turns into \(x^2 - 8x + 16\), a perfect square trinomial. This method helps in systematically transforming any quadratic into a more manageable form, making it easier to factor.

Binomial

A binomial is a polynomial with exactly two terms. It is one of the simplest forms of polynomial expressions and lays the foundation for understanding more complex expressions like trinomials.

In our exercise, \(x^2 - 8x\) represents a binomial. Here, \(x^2\) and \(-8x\) are the two terms that make up the binomial. Recognizing and working with binomials is crucial because they often act as starting points for operations like factoring and completing the square.

Binomials can easily be expanded into trinomials by adding another term. Understanding how to manipulate binomials is key for solving many algebraic problems, especially when dealing with quadratic expressions.

Trinomial

A trinomial is a polynomial expression containing three terms. An example is the expression \(x^2 - 8x + 16\), which is the result of transforming the binomial \(x^2 - 8x\) using the method of completing the square.

Trinomials are significant in algebra as they frequently appear in quadratic equations, and recognizing a perfect square trinomial can simplify the process of solving these equations. A perfect square trinomial can always be factored into the square of a binomial, which is clearly demonstrated in our exercise where \(x^2 - 8x + 16\) is factored into \((x - 4)^2\).

Understanding trinomials and their properties helps in recognizing patterns and applying suitable algebraic methods for simplification and solution.