Problem 45 - Algebra 1: Concepts and Skills

Question

$$ (5 t-3)(2 t+3) $$

Step-by-Step Solution

Answer

The result of the multiplication is \(10 t^2 + 9 t - 9\).

Step 1: Applying FOIL Method - First Terms

Multiply the first terms of both binomials together. In this case, \(5 t * 2 t\) which equals \(10 t^2\).

Step 2: Applying FOIL Method - Outer Terms

Multiply the outer terms together. In this case, \(5 t * 3\) which equals \(15 t\).

Step 3: Applying FOIL Method - Inner Terms

Multiply the inner terms together. In this case, \(-3 * 2 t\) which equals \(-6 t\).

Step 4: Applying FOIL Method - Last Terms

Multiply the last terms of both binomials together. In this case, \(-3 * 3\) which equals \(-9\).

Step 5: Combine like terms

The last step is to combine like terms from the expressions calculated in the steps 2 and 3: \(15 t - 6 t = 9 t\). Therefore, the expression would be \(10 t^2 + 9 t - 9\).

Key Concepts

Understanding Binomials

A binomial is a polynomial with just two terms. These terms are typically connected by a plus or minus sign. For instance, in the expression \((5t - 3)(2t + 3)\), both \(5t - 3\) and \(2t + 3\) are binomials.

Binomials can take on a variety of forms, such as:

• Identical terms: where both terms contain the same variable, like \(x + y\)
• Mixed terms: where terms may contain different variables or constants, like \(3x + 5\)

The general structure of a binomial is \(a + b\) or \(a - b\), where \(a\) and \(b\) can be numbers, variables, or a combination of both. In our exercise, we work with variables accompanied by coefficients.
The FOIL method is particularly useful when multiplying binomials, as it helps ensure that each term is correctly accounted for during multiplication. Once you recognize a binomial, applying the FOIL method becomes much easier.

Exploring Polynomial Multiplication

Polynomial multiplication involves combining polynomials through multiplication, resulting in a new polynomial. When multiplying binomials, the process is simplified by using the FOIL method, which stands for First, Outer, Inner, Last.

Here's how it works:

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

Let’s apply these to our example \((5t - 3)(2t + 3)\):

• First: \(5t \times 2t = 10t^2\)
• Outer: \(5t \times 3 = 15t\)
• Inner: \(-3 \times 2t = -6t\)
• Last: \(-3 \times 3 = -9\)

Each of these multiplicative steps ensures that components from both binomials are included in the final expression.
In this manner, the multiplication of two binomials becomes manageable by breaking it into smaller, easier steps.

Combining Like Terms

Once you've multiplied polynomials, combining like terms comes next. Like terms are terms that have identical variable parts. This means they have the same variable raised to the same power.

For example, in the expression \(15t - 6t\), both terms are like terms because they both contain the variable \(t\). By combining these, we simplify the expression:

• \(15t - 6t\) simplifies to \(9t\).

Combining like terms is crucial in simplifying polynomial expressions post-multiplication. It reduces the expression to its simplest form and makes it more readable.
In our complete expression \(10t^2 + 15t - 6t - 9\), combining the like terms \(15t\) and \(-6t\) gives us \(9t\). Thus, our simplified result is \(10t^2 + 9t - 9\).
Regular practice in identifying and combining like terms builds familiarity and confidence in working with polynomials.