Problem 25 - Algebra 2 and Trigonometry

Question

In \(9-26,\) write each expression as the product of two binomials. $$ 4 y^{2}+4 y+1 $$

Step-by-Step Solution

Answer

The expression is \((2y + 1)^2\).

Step 1: Identify the Quadratic Form

Recognize that the expression \(4y^2 + 4y + 1\) is a quadratic trinomial and can be written in the form \(ax^2 + bx + c\), where \(a = 4\), \(b = 4\), and \(c = 1\). The task is to rewrite this expression as the product of two binomials.

Step 2: Check for Perfect Square Trinomial

Notice whether the quadratic trinomial can be a perfect square. A trinomial \(ax^2 + bx + c\) is a perfect square if it can be expressed as \((mx + n)^2 = m^2x^2 + 2mnx + n^2\). Here, verify whether \(4y^2 + 4y + 1\) can be rewritten in this form.

Step 3: Compare with Perfect Square Formula

Compare \(4y^2 + 4y + 1\) to the expanded form of a perfect square \((2y + 1)^2 = (2y)^2 + 2(2y)(1) + 1^2\). Calculate each part: \((2y)^2 = 4y^2\), \(2(2y)(1) = 4y\), and \(1^2 = 1\). Our trinomial matches the components, confirming it's a perfect square.

Step 4: Express as Product of Binomials

Since \(4y^2 + 4y + 1\) is a perfect square trinomial, it can be expressed as \((2y + 1)(2y + 1)\) or simply \((2y + 1)^2\). This is rewritten as a product of two identical binomials.

Key Concepts

Perfect Square Trinomial

If you're working with quadratic expressions, identifying a perfect square trinomial is crucial. A perfect square trinomial takes the form

• \(ax^2 + bx + c\)

And can be expressed as a square of a binomial. Consider

• \((mx + n)^2 = m^2x^2 + 2mnx + n^2\)

In simpler terms, when a trinomial fits this pattern, all parts can be matched exactly. Let's apply it to the expression given:

1. \(4y^2\) corresponds to \((2y)^2\)
2. \(4y\) coincides with \(2(2y)(1)\)
3. \(1\) is exactly \(1^2\)

Since these match completely, our expression is indeed a perfect square trinomial. This is key to simplifying and factoring expressions effortlessly.

Binomial Product

Transforming a perfect square trinomial into a binomial product is straightforward when you understand the pattern. The expression

• \(4y^2 + 4y + 1\)

is rewritten through the lens of a binomial. We take the square root of the leading term and the constant term, which are:

1. The square root of \(4y^2\) is \(2y\)
2. The square root of \(1\) is \(1\)

So, our binomial becomes:

• \(2y + 1\)

Now express this binomial squared:

• \((2y + 1)^2 = (2y+1)(2y+1)\)

By identifying the binomial, you can visualize and expand quickly if needed. This factorization technique ensures that complex trinomials become manageable.

Quadratic Trinomial

Quadratic trinomials often intimidate students, but simplifying them doesn't have to be difficult. To start, recognize that a quadratic trinomial is any polynomial of degree two in the form:

• \(ax^2 + bx + c\)

Each term has a specific role:

1. \(a\) is the coefficient of the quadratic term
2. \(b\) adjusts the linear term
3. \(c\) represents the constant

The goal is to restructure it into a product of binomials. In our case,

• \(4y^2 + 4y + 1\)

represents

• \(a = 4\), \(b = 4\), and \(c = 1\)

We identified it as a perfect square, simplifying our task. By squaring the identified binomial

• \(2y + 1\)

we achieve a concise factorized form. Grasping this concept allows you to handle many algebraic expressions effectively, providing a strong foundation in algebra.