Problem 19 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 2+24 y+40 y^{2} $$

Step-by-Step Solution

Answer

The expression factors to \(2(10y+1)(2y+1)\).

Step 1: Identify the Expression

The given expression is \(2 + 24y + 40y^2\). We need to factor it completely.

Step 2: Factor Out the Greatest Common Factor (GCF)

First, identify the greatest common factor of all terms in the expression. The coefficients are 2, 24, and 40. The GCF is 2. Factor out 2 from the entire expression:\[2(1 + 12y + 20y^2)\]

Step 3: Arrange the Remaining Expression

Rewrite the expression inside the parentheses in a standard quadratic form: \[20y^2 + 12y + 1\]

Step 4: Factor the Quadratic Expression

Attempt to factor the quadratic expression \(20y^2 + 12y + 1\) as a product of two binomials. We are looking for two numbers whose product is \(20 \times 1 = 20\) and whose sum is 12.

Step 5: Find Suitable Pair of Factors

The pair of numbers that multiplies to 20 and adds to 12 are 2 and 10. Rewrite the quadratic expression using these factors: \[20y^2 + 2y + 10y + 1\]

Step 6: Group Terms and Factor by Common Terms

Group the expression to factor by pairs: \[(20y^2 + 10y) + (2y + 1)\]Factor each group separately:\[10y(2y + 1) + 1(2y + 1)\]

Step 7: Combine Like Terms

Since \(2y + 1\) is common in both groups, factor it out:\[(10y + 1)(2y + 1)\]

Step 8: Return to the Original Factored Expression

Since we factored out a 2 at the start, the fully factored form of the original expression is:\[2(10y + 1)(2y + 1)\]

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is the largest number that can divide two or more numbers evenly.
It is one of the most basic concepts in factoring. To find the GCF of a set of numbers, you need to identify the largest factor that each number shares. Here's how you can do it:

• List the factors of each number involved.
• Identify the largest factor that appears in every list.

In our expression, we have the coefficients 2, 24, and 40. By listing their factors:

• Factors of 2: 1, 2
• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40

The greatest common factor, which is common in all these lists, is 2.
You factor this out from the entire expression to simplify the problem.

Quadratic Expression

A quadratic expression is an algebraic expression of the form ax² + bx + c, where a, b, and c are constants, and x is the variable.
This type of expression is characterized by having the highest degree of 2. Usually, the goal is to factor these expressions into simpler ones, often into two binomial expressions.
When factoring, the key is to identify two numbers that multiply to the product of the a and c terms (in a location like ax² + bx + c) and add up to the b term.For example, in the simplified quadratic expression after factoring the GCF from our original problem, we had to factor:\[20y^2 + 12y + 1\]Here:

• The product of the coefficient of y² (20) and the constant term (1) is 20.
• We needed to find two numbers that multiplied to 20 and added to the middle coefficient, 12.

This crucial step allows us to set the stage for breaking down a quadratic into simpler forms that can then be easily solved or used in solving larger polynomial equations.

Binomials

A binomial is a polynomial with exactly two terms.
When factoring quadratic expressions, you'll frequently aim to express them as the product of two binomials.For instance, after factoring out the GCF and rewriting our quadratic expression, the expression:\[20y^2 + 2y + 10y + 1\]was factored to group terms and ultimately rewritten as:\[(2y + 1)\]and\[(10y + 1)\]These are binomials because they consist of two terms each.
When they are multiplied back (using the distributive property), they recreate the original quadratic expression. The process of factoring into binomials not only simplifies the expression, but it is also essential for solving quadratic equations as it leads to setting each factor to zero and solving for the variable. This makes binomials a vital concept in algebra.