Problem 3 - Elementary and Intermediate Algebra

Question

The first terms of the binomial factors \((5 y+1)(y+3)\) are _____ and _____. The second terms of the binomial factors are _____ and _____.

Step-by-Step Solution

Answer

First terms: 5y and y. Second terms: 1 and 3.

Step 1: Identify the First Terms

In the binomial expression \[(5y+1)(y+3),\]the first term of each factor is simply the term that appears before any addition or subtraction. Look at both binomial factors.- For the first factor, \((5y + 1)\), the first term is \(5y\).- For the second factor, \((y + 3)\), the first term is \(y\). Thus, the first terms of the binomial factors are \(5y\) and \(y\).

Step 2: Identify the Second Terms

The second term in each binomial factor is the term that follows the addition symbol.- In the first binomial \((5y+1)\), the second term is \(1\).- In the second binomial \((y+3)\), the second term is \(3\). Thus, the second terms of the binomial factors are \(1\) and \(3\).

Key Concepts

Binomial Factors

Binomial factors are expressions consisting of two terms separated by a plus or minus sign. Each part of these factors is a crucial component, as they determine the result when multiplied together. In algebra, understanding binomial factors helps simplify expressions and solve equations more efficiently. These factors are commonly seen in problems involving the distributive property and polynomial expansions.

• The binomial expression \((5y+1)(y+3)\) is an example, where each set of parentheses holds a binomial factor.
• Inside these parentheses, each term interacts with its counterpart from other factors during multiplication.
• These factors aren't just limited to numbers; they can represent variables, coefficients, or both.

Recognizing binomial factors is the first step in various algebraic manipulations and provides the basis for further operations.

First and Second Terms

In each binomial factor, there are two distinct parts: the first term and the second term. Identifying these terms is essential in handling algebraic expressions effectively.

• For the binomial \((5y + 1)\), the first term is \(5y\), the part present before any addition or subtraction.
• For the binomial \((y + 3)\), the first term is \(y\).

Similarly, understanding the second terms:

• In \((5y + 1)\), the second term following the addition is \(1\).
• In \((y + 3)\), the second term is \(3\).

Recognizing and distinguishing between these first and second terms is fundamental for multiplying binomials, as it determines how terms will interact with each other.

Multiplication of Binomials

Multiplying binomials involves distributing each term in the first binomial to every term in the second binomial. This process is often remembered as FOIL, which stands for:

• First: Multiply the first terms from each binomial.
• Outer: Multiply the outer terms from each binomial.
• Inner: Multiply the inner terms from each binomial.
• Last: Multiply the last terms from each binomial.

For \((5y+1)(y+3)\), the multiplication process will unfold as follows:

• First: \(5y \times y = 5y^{2}\)
• Outer: \(5y \times 3 = 15y\)
• Inner: \(1 \times y = y\)
• Last: \(1 \times 3 = 3\)

Finally, sum up all the products to yield the expression:\(5y^{2} + 15y + y + 3\).
Combining like terms gives you: \(5y^{2} + 16y + 3\).
Understanding the multiplication of binomials is crucial in expanding expressions and solving polynomial equations.