Problem 94 - Algebra 1

Question

Factor the trinomial. $$x^{2}+18 x+81$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(x^{2}+18x+81\) is \((x + 9)^2\)

Step 1: Identify a Perfect Square Trinomial

Recognize the trinomial as a perfect square trinomial. This is done by comparing the given trinomial, \(x^{2}+18x+81\) with the perfect square trinomial pattern \(a^2 + 2ab + b^2\). In this case, \(a=x\), \(2ab=18x\), and \(b^2=81\).

Step 2: Find the square root of first and last term

The square root of first term \(a^2\) is \(a\) which is \(x\). The square root of last term \(b^2\) is \(b\), which is \(9\), because \(9 * 9 = 81\). We use these values to verify if \(2ab = 18x\).

Step 3: Confirm 2ab = the middle term

This verifies that \(2*a*b = 2*x*9 =18x\) which is indeed the middle term of the trinomial.

Step 4: Express as a square of a binomial

Since all conditions for a perfect square trinomial are fulfilled, the trinomial can be expressed as a square of a binomial: \( (a + b) ^ 2 = (x + 9) ^ 2\)

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a specific form of a polynomial characterized by its structure, which is always expressible as the square of a binomial. The typical form of a perfect square trinomial is given by the expression \[a^2 + 2ab + b^2.\]
In this structure:

• The first term \(a^2\) and the third term \(b^2\) are perfect squares themselves.
• The second term \(2ab\) is exactly twice the product of the square roots of the first and third terms.

For example, consider the trinomial \(x^2 + 18x + 81\). Comparing this with the pattern \(a^2 + 2ab + b^2\), we identify that \(a = x\), and the middle term \(18x\) confirms that \(b = 9\). This structure is crucial when factoring polynomials as it allows us to simplify expressions efficiently.

Binomial

A binomial is any algebraic expression that consists of exactly two distinct terms. For instance, \(x + y\) and \(3a - 4b\) are both binomials. They may include variables, numbers, or both.
Binomials are a fundamental component of algebra and are frequently encountered in various operations:

• They are the building blocks for more complex expressions, like trinomials and polynomials.
• Binomials follow specific patterns when multiplied, as seen in the formula \((a + b)^2 = a^2 + 2ab + b^2\).
• They can be factored and expanded using different algebraic identities.

Understanding binomials is key to mastering algebraic factorizations. In the context of our exercise, recognizing the binomial \(x + 9\) makes the factoring process much smoother.

Squared Binomial

A squared binomial refers to the product that results when a binomial is multiplied by itself. It takes the general form of \((a + b)^2\) which can be expanded or simplified to \(a^2 + 2ab + b^2\).
This concept is critical in the factorization process, especially when working with quadratic expressions.

• The squared form provides both a shortcut for multiplication and an insight into the structure of perfect square trinomials.
• Recognizing this pattern helps to quickly factor trinomials of the form \((a + b)^2\) into \((a + b)^2\), or the reverse process.
• It streamlines solving quadratic equations and simplifying polynomial expressions.

In our exercise, once the trinomial \(x^2 + 18x + 81\) is identified as a perfect square trinomial, we can quickly ascertain that it is the square of \(x + 9\). This association simplifies the problem-solving process, demonstrating the elegance and utility of squared binomials in algebra.