Problem 9 - Elementary Algebra

Question

For the following problems, factor, if possible, the polynomials. $$ 9 y^{2}-30 y+25 $$

Step-by-Step Solution

Answer

Answer: The factored form of the given quadratic polynomial is $$(3y - 5)^2$$.

Step 1: Determine if the polynomial is a perfect square trinomial

Let's check if the given polynomial $$9y^2 - 30y + 25$$ can be written as a perfect square trinomial. A perfect square trinomial has the form $$(ay + b)^2 = a^2y^2 + 2aby + b^2$$ Comparing the coefficients, we can see that: - $$a^2 = 9 \Rightarrow a = \pm3$$ - $$b^2 = 25 \Rightarrow b = \pm5$$ - $$2ab = -30 \Rightarrow ab = -15$$ From the above equations, we can choose $$a = 3$$ and $$b = -5$$ since their product is -15. Therefore, the given polynomial can be written as $$(3y - 5)^2$$.

Step 2: Factor the polynomial

As we determined in step 1 that the given polynomial is a perfect square trinomial, we can factor it as: $$9y^2 - 30y + 25 = (3y - 5)^2$$

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of trinomial that can be expressed as the square of a binomial. This means it has the form \((ay + b)^2 = a^2y^2 + 2aby + b^2\). Recognizing a perfect square trinomial involves checking if the first and last terms are perfect squares and the middle term is twice the product of the square roots of these terms.
When factoring a trinomial, like \(9y^2 - 30y + 25\), one must check if it follows this pattern. In this case:- The first term \(9y^2\) is the square of \(3y\).- The last term \(25\) is the square of \(-5\).- The middle term \(-30y\) is twice \(3\) and \(-5\), or \(2 \, \times \, 3 \, \times \, -5\).
By checking these conditions, we identify that the trinomial is indeed a perfect square, and can be factored as \((3y - 5)^2\). Recognizing these patterns simplifies the process of polynomial factoring.

Coefficients

Coefficients in a polynomial are crucial as they determine the numerical characteristics of each term in the expression. In a quadratic expression like \(9y^2 - 30y + 25\), the coefficients are the numbers multiplying the variables:

• The coefficient of \(y^2\) is \(9\).
• The coefficient of \(y\) is \(-30\).
• The constant term is \(25\).

Understanding coefficients helps in identifying patterns, especially when checking for perfect square trinomials or factoring. For instance:
- Verify if the leading coefficient \(a^2\) matches a perfect square, and compare it to the related term.- The middle term's coefficient informs if it fits into the \(2ab\) section, which confirms if a trinomial can be decomposed into a binomial squared.

Polynomial Factoring

Polynomial factoring is a method used to express a polynomial as a product of its simpler factors. It involves breaking down expressions to their core multiplicative components. In the expression \(9y^2 - 30y + 25\), we factored it into \((3y - 5)^2\), a simpler expression that, when expanded, returns the original statement.
Factoring can involve different techniques depending on the structure of the polynomial:

• Finding GCF: Identify the greatest common factor for every term.
• Perfect Square Trinomial: Recognize patterns using squared binomials.
• Difference of Squares: Use if the polynomial takes a(a - b)(a + b) form.
• Grouping: For polynomials with more than three terms.

The choice of method often simplifies complex expressions and aids in solving equations or evaluating polynomials efficiently.

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree two, typically formatted as \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). For the expression \(9y^2 - 30y + 25\):

• \(a = 9\)
• \(b = -30\)
• \(c = 25\)

Quadratic expressions can often be factored into products involving binomials, especially when they form perfect square trinomials as seen in this example.
The ability to recognize and manipulate quadratics is fundamental in algebra, as it extends to solving quadratic equations, graphing quadratic functions, and modeling real-world scenarios. Understanding their standard format helps identify the path to simplification through methods like factoring, completing the square, or using the quadratic formula. The focus should always be on aligning theoretical concepts with practical solving techniques.