Problem 24 - Precalculus Mathematics for Calculus

Question

Multiply the algebraic expressions using the FOIL method and simplify. $$(4 s-1)(2 s+5)$$

Step-by-Step Solution

Answer

The simplified product is \(8s^2 + 18s - 5\).

Step 1: Identify the binomials

In the expression \((4s - 1)(2s + 5)\), we need to recognize the two binomials: the first binomial is \(4s - 1\) and the second binomial is \(2s + 5\). We are going to apply the FOIL method, which stands for First, Outer, Inner, Last.

Step 2: Multiply the 'First' terms

The 'First' terms in the binomials are \(4s\) from \((4s - 1)\) and \(2s\) from \((2s + 5)\). Multiply these terms: \[ 4s \times 2s = 8s^2 \].

Step 3: Multiply the 'Outer' terms

The 'Outer' terms are \(4s\) from \((4s - 1)\) and \(5\) from \((2s + 5)\). Multiply these terms: \[ 4s \times 5 = 20s \].

Step 4: Multiply the 'Inner' terms

The 'Inner' terms are the \(-1\) from \((4s - 1)\) and \(2s\) from \((2s + 5)\). Multiply these terms:\[ -1 \times 2s = -2s \].

Step 5: Multiply the 'Last' terms

The 'Last' terms are \(-1\) from \((4s - 1)\) and \(5\) from \((2s + 5)\). Multiply these terms:\[ -1 \times 5 = -5 \].

Step 6: Combine all the products

Now, add all the products obtained from Steps 2 to 5:\[ 8s^2 + 20s - 2s - 5 \].

Step 7: Simplify the expression

Combine like terms (the terms involving \(s\)) in the expression:\[ 8s^2 + (20s - 2s) - 5 = 8s^2 + 18s - 5 \].This is the simplified form of the expression.

Key Concepts

Binomials

When dealing with polynomial multiplication, a fundamental component to understand is binomials. A binomial is an algebraic expression that consists of two terms connected by either a plus or a minus sign. In the expression \((4s - 1)(2s + 5)\), there are two binomials: \(4s - 1\) and \(2s + 5\). These binomials are composed of variables and coefficients, each representing a single term.

• The first binomial \(4s - 1\) has terms \(4s\) and \(-1\).
• The second binomial \(2s + 5\) includes \(2s\) and \(5\).

Recognizing these builds the foundation for using the FOIL method, a technique specific to multiplying two binomials.

Distributive Property

The distributive property is a crucial concept in mathematics that allows us to multiply a single term with two or more terms inside a parenthesis. This property states that multiplying a number by a sum of numbers is the same as doing each multiplication separately.
In polynomial multiplication, especially when using the FOIL method, each term in one binomial is "distributed" or multiplied with each term in the other binomial. This step-by-step method ensures all combinations of terms are accounted for. This is how we approach the multiplication in the expression \((4s - 1)(2s + 5)\).
Here's how it aligns with the FOIL process:

• First, we take the first terms from both binomials and multiply them together.
• Next, we handle the outer terms, which are the first term from the first binomial and the last term from the second binomial.
• Then, the inner terms are multiplied, which involves the last term of the first binomial and the first term of the second.
• Finally, we multiply the last terms of both binomials together.

By applying the distributive property, we efficiently reach the final multiplied expression.

Polynomial Multiplication

Polynomial multiplication involves expanding expressions with multiple terms and combining like terms for simplification. Through the FOIL method, a specific technique tailored for binomials, we systematically multiply the terms in the expression \((4s - 1)(2s + 5)\).
This technique helps in:

• Breaking down the process into manageable steps: First, Outer, Inner, and Last.
• Ensuring all pairs of terms are multiplied between the binomials.

Once each product is computed, they are combined to form a single expression:

• \(8s^2\): Resulting from multiplying the first terms.
• \(20s\) from the outer terms, and \(-2s\) from the inner terms which combine to form \(18s\).
• Finally, \(-5\) from the last terms.

This organized approach simplifies potentially complex polynomial multiplications, leading to the simplified result of \(8s^2 + 18s - 5\). Understanding this process is very beneficial in handling more complex algebraic expressions efficiently.