Problem 33 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. \(y^{2}-7 y+5\)

Step-by-Step Solution

Answer

The factored form of \(y^{2}-7 y+5\) is \((y-2)(y-5)\)

Step 1: Identify the coefficients

The coefficients in the trinomial \(y^{2}-7 y+5\) are a=1 (coefficient of \(y^{2}\)), b=-7 (coefficient of y), and c=5 (constant term).

Step 2: Find the Factors

Look for two numbers that multiply to 5 (a*c) and add to -7 (b). Here those numbers are -2 and -5 because -2*-5=5 and -2 - 5=-7.

Step 3: Write the Factorization

The trinomial \(y^{2}-7 y+5\) can be factored as \((y-2)(y-5)\).

Step 4: Check Factorization Using FOIL

To check the answer, expand \((y-2)(y-5)\) using FOIL: first terms y*y=\(y^{2}\), outside terms y*-5=-5y, inside terms -2*y=-2y, and last terms -2*-5=5. Add these up to get \(y^{2}\) -5y -2y+5 = \(y^{2}\) -7y +5, which matches the original expression, confirming that the factorization is correct.

Key Concepts

Coefficients Identification

Identifying coefficients in a trinomial is a crucial first step in the process of factoring. The coefficient is essentially a number placed in front of a variable within an expression, indicating how many times that variable is being considered. In mathematical terms, for a general trinomial expression of the form \(ax^2 + bx + c\), the coefficients are simply \(a\), \(b\), and \(c\).
In the expression \(y^2 - 7y + 5\), the coefficients are identified as follows:

• \(a = 1\), which is the coefficient of \(y^2\)
• \(b = -7\), which is the coefficient of \(y\)
• \(c = 5\), which is the constant term

Knowing these coefficients helps us determine the pair of numbers that is needed to factor the trinomial effectively. This simplifies the path toward a successful factorization since choosing the correct pair of numbers relies heavily on understanding the coefficients.

FOIL Multiplication

The FOIL method is a process used to multiply two binomials. It's a mnemonic that stands for First, Outside, Inside, Last, referring to the position of each term when the multiplication is carried out. By applying FOIL, we can ensure that all parts of the binomials are multiplied correctly to form the expanded expression.
When you want to check your factorization, as with changing \((y-2)(y-5)\) back to its original expanded form of \(y^2 - 7y + 5\), the FOIL method is particularly useful:

• First: Multiply the first terms: \(y \cdot y = y^2\)
• Outside: Multiply the outer terms: \(y \cdot -5 = -5y\)
• Inside: Multiply the inner terms: \(-2 \cdot y = -2y\)
• Last: Multiply the last terms: \(-2 \cdot -5 = 10\)

Add all these results together to get \(y^2 - 5y - 2y + 10 = y^2 - 7y + 5\). Using FOIL helps confirm whether the factorization is done correctly and is the essence of both verifying and solidifying understanding of multiplication of binomials.

Polynomial Expressions

Polynomial expressions are algebraic expressions that include several terms, each consisting of variables raised to a power and multiplied by coefficients. These can range from simple binomials to complex trinomials and beyond. Understanding polynomials is essential because they appear frequently in various areas of mathematics.
A trinomial is a type of polynomial expression with exactly three terms. Specifically, in the expression \(y^2 - 7y + 5\), we observe:

• A quadratic term \(y^2\)
• A linear term \(-7y\)
• A constant term \(5\)

When factoring trinomials, our goal is to rewrite them as a product of two binomials, if possible. This process relies on effectively identifying patterns in quadratic expressions and applying methods such as completing the square or using the quadratic formula, especially when the trinomial cannot easily be factored by simple inspection. However, as with our initial problem, some trinomials can be split into easily manageable components through basic factorization, aiding in simplifying and solving equations involving these expressions.