Problem 27 - College Algebra

Question

Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. $$ x^{2}+12 x $$

Step-by-Step Solution

Answer

The constant to be added is 36. The factored form of the trinomial is \((x+6)^{2}\).

Step 1: Identify Values

From the binomial \(x^{2}+12 x\), we can identify \(a\) as \(x\) and \(b\) as \(6\) because \(2ab = 12x\). Each term in the binomial represents the terms \(a^{2}\) and \(2ab\) respectively in the perfect square trinomial formula \(a^{2}+2ab+b^{2}\).

Step 2: Determine the Constant

Now we can calculate the constant to be added to make the binomial a perfect square trinomial. Using the formula \(a^{2}+2ab+b^{2}\), the constant is the term \(b^{2}\), which is \(6^{2} = 36\).

Step 3: Write and Factor the Trinomial

The required perfect square trinomial is \(x^{2} + 12x + 36\). To factor, we can write it in the form of \((a+b)^{2}\) which is \((x+6)^{2}\).

Key Concepts

Factoring

Factoring is a fundamental concept in algebra that involves breaking down expressions into simpler components, usually in the form of a product of their factors. It is like reverse expansion. When you factor a polynomial, you look for expressions that, when multiplied together, produce the original polynomial.
Let's consider the trinomial we derived: \(x^2 + 12x + 36\). To factor this, we aim to express it as a product of binomials. Since it's a perfect square trinomial, it simplifies as \((x+6)^2\).
This factoring tells us that the expression is made by squaring the binomial \((x+6)\).

• Factoring simplifies expressions.
• Aids in solving equations.
• Helps identify roots or solutions.

Understanding factoring is crucial to solving quadratic equations and analyzing functions efficiently.

Binomial

A binomial is a polynomial with exactly two terms. These two terms are connected by a sum or a difference. In our exercise, the binomial is \(x^2 + 12x\). This is an algebraic expression containing two distinct terms that can be combined using specific operations.
The two terms can often be identified as parts of a larger polynomial.

• A binomial consists of two terms.
• It's a building block in polynomials.
• Can be part of a trinomial or larger polynomial.

Our task often involves adding the right constant to transform this binomial into a more structured form like a perfect square trinomial, which can then be easily factored.

Constant Addition

The method of constant addition involves adding a specific value to a polynomial to transform it into another mathematical expression, often to create a desired form. In this exercise, we aim to add a constant to the binomial \(x^2 + 12x\) to make it a perfect square trinomial.
By identifying the expression \(2ab = 12x\), we determine \(b = 6\), and thus the constant to add is \(6^2 = 36\).
This transforms the expression into \(x^2 + 12x + 36\), a perfect square trinomial.

• Identifies the transformation needed.
• Helps achieve a specific polynomial form.
• Aids in easier factoring or solving.

Adding the right constant simplifies solving polynomials by shaping them into recognizable patterns.

Trinomial Formula

The trinomial formula is a powerful tool in algebra for identifying and working with perfect square trinomials. A perfect square trinomial takes the form \(a^2 + 2ab + b^2\), which can be rewritten as \((a+b)^2\).
In our example, after adding the correct constant to \(x^2 + 12x\), it becomes \(x^2 + 12x + 36\), following this pattern. It is factored as \((x+6)^2\).
Following this formula:

• \(a^2\) corresponds to the square of the first term.
• \(2ab\) represents twice the product of two specific terms.
• \(b^2\) is the square of the constant added.

Using the trinomial formula aids in swiftly converting binomials to perfect square trinomials, facilitating their factorization and solving related quadratic equations smoothly.