Problem 16 - Introductory Algebra for College Students

Question

Use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$5 y^{2}-8 y+3$$

Step-by-Step Solution

Answer

The factorised form of the trinomial \(5y^{2}-8y+3\) is \((5y - 5)(y - 1)\).

Step 1: Identify the Numbers

Identify two numbers that multiply to 15 and add up to -8. These numbers are -5 and -3 because \(-5 \times -3 = 15\) and \(-5 - 3 = -8\).

Step 2: Split the Middle Term

Split the middle term of the trinomial using the numbers found in step 1. The trinomial becomes \(5y^{2}-5y-3y+3\).

Step 3: Factor by Grouping

Factor the trinomial by grouping, this involves factorising the first two terms and the last two terms separately. This gives \(y(5y-5)-1(3y-3)\).

Step 4: Factorise Further

Further factorise the expression if required. By factoring out \(5y - 5\) from the terms, the fully factorised form will be \((5y-5)(y-1)\).

Step 5: Check via FOIL Multiplication

Verify the factorisation by applying FOIL (First - Outer - Inner - Last) multiplication. \((5y - 5)(y - 1)\) expands to \(5y^{2}-8y+3\) through FOIL multiplication.

Key Concepts

FOIL multiplication

FOIL multiplication is a handy technique for multiplying two binomials, and it incorporates four steps indicated by the acronym FOIL: First, Outer, Inner, and Last.

Let's take the general case of multiplying binomials, \( (a+b)(c+d) \). To use FOIL, multiply the First terms in each binomial \( (a \times c) \), the Outer terms \( (a \times d) \), the Inner terms \( (b \times c) \), and finally the Last terms \( (b \times d) \). After performing these multiplications, you add the results together to get the expanded form \( ac + ad + bc + bd \).

In the context of checking factorisations, as with our trinomial \( 5y^2 - 8y + 3 \), FOIL helps verify that the factored form, when multiplied, returns to the original trinomial. This reassurance is crucial as it ensures that the process of factorisation has been done correctly.

Factor by grouping

Factor by grouping is a technique used when dealing with polynomials with four or more terms. The method involves organizing the terms into groups and factoring out common factors from each group separately.

Here's how it works: Take the partially factored trinomial \( 5y^2-5y-3y+3 \). You would divide this expression into two groups: \( (5y^2-5y) \text{and}\ (-3y+3) \). Next, factor out the greatest common factor (GCF) from each group, leaving you with \( y(5y-5) \text{and}\ -1(3y-3) \). If done correctly, this method will often reveal a common binomial factor. In this case, both groups contain the binomial \( 5y-5 \), and factoring that out gives you the fully factored form of the original trinomial: \( (5y-5)(y-1) \). This strategy greatly simplifies some otherwise intimidating algebra problems.

Factorisation in algebra

Factorisation is a foundational concept in algebra that involves breaking down a complex expression into a product of simpler factors. This process is crucial when solving equations, simplifying expressions, or finding roots of polynomials.

Factoring trinomials, a specific case of factorisation, requires identifying a pair of numbers that satisfy specific conditions related to the trinomial's coefficients. For example, for a trinomial like \( 5y^2-8y+3 \) the two numbers must multiply to the product of the coefficient of \( y^2 \), which is 5 in this case, and the constant term, here 3; hence, \(5 \times 3 = 15\). These numbers also must add up to the coefficient of \( y \), which is -8.

Finding the right pair of numbers is sometimes trial and error, but with practice, one develops an intuition for spotting patterns and identifying these pairs more efficiently. Once the correct numbers are determined, factoring trinomials can become a much more straightforward task.