Problem 74 - Algebra for College Students

Question

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$9 a^{2}-30 a+25$$

Step-by-Step Solution

Answer

The polynomial factors to \((3a - 5)^2\), a perfect square trinomial.

Step 1: Recognize the Structure

The given polynomial is a quadratic expression in the form of \(ax^2 + bx + c\). It is \(9a^2 - 30a + 25\), which resembles the structure of a perfect square trinomial.

Step 2: Check for a Perfect Square Trinomial

In a perfect square trinomial, the first and last terms are perfect squares, and the middle term is twice the product of the square roots of these terms. First term: \(9a^2 = (3a)^2\), last term: \(25 = 5^2\). Check middle term: \(-30a = 2 \times (3a) \times 5\). The middle term matches, so it's a perfect square: \((3a - 5)^2\).

Step 3: Confirm the Factorization

Confirm by expanding \((3a - 5)^2\): \((3a - 5)(3a - 5) = 9a^2 - 15a - 15a + 25 = 9a^2 - 30a + 25\). The expansion confirms the expression \((3a - 5)^2\) equals the original polynomial.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special kind of quadratic polynomial. It forms when a binomial is multiplied by itself – essentially, it's a squared binomial. You'll recognize it by checking if the first and last terms of the trinomial are perfect squares. For example, in the polynomial \(9a^2 - 30a + 25\), the first term, \(9a^2\), is the square of \((3a)\) and the last term, \(25\), is the square of \(5\).
The crucial part is the middle term. It should be twice the product of the roots of the first and last terms. Essentially, the middle term \(-30a\) should equal \(-2 imes (3a) imes 5\). It fits this criteria, confirming it's a perfect square trinomial.
Once identified, it can be factored into \((3a - 5)^2\), as each term essentially reflects the structure of a squared binomial.

Quadratic Expression

Quadratic expressions are polynomials of degree two, which means they can be written in form \(ax^2 + bx + c\). They are essential in algebra due to their wide applicability in various fields, including physics and engineering.
In our exercise, the expression \(9a^2 - 30a + 25\) fits this form, with the values of \(a = 9\), \(b = -30\), and \(c = 25\). Understanding this structure is vital for recognizing patterns like those found in perfect square trinomials and applying factorization techniques effectively.
These expressions often model real-world situations, ranging from projectile motion to area problems, making learning to manage them a key math skill.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into simpler, non-divisible polynomials whose product is the original polynomial. It's comparable to prime factorization with integers.
The significance of factorization lies in simplifying polynomials for easier manipulation and solution finding, crucial in solving quadratic equations efficiently.
For the exercise \(9a^2 - 30a + 25\), we identify it as a perfect square trinomial and factor it as \((3a - 5)^2\). Factorization can appear in different forms, such as grouping, and recognizing these forms helps in simplifying expressions.

Algebraic Techniques

Algebraic techniques involve a variety of methods used to manipulate and solve polynomial equations. These include methods for recognizing patterns like the perfect square trinomial, using formulas like the quadratic formula, and employing strategies like completing the square.
Recognizing patterns allows us to factor quickly; for example, identifying a perfect square trinomial quickly narrows the approach. We used this approach to transform \(9a^2 - 30a + 25\) into \((3a - 5)^2\).
Additionally, these techniques help manage more complicated equations by breaking them down into simpler parts, which are easier to solve or evaluate. Mastery of algebraic techniques ensures efficiency in solving diverse mathematical problems.