Problem 70 - Algebra A Combined Function

Question

Multiply. \((t+3)\left(t^{2}-5 t+5\right)\)

Step-by-Step Solution

Answer

The product is \(t^3 - 2t^2 - 10t + 15\).

Step 1: Identify the Expression Form

We are given the expression \((t + 3)(t^2 - 5t + 5)\), which consists of a binomial \((t + 3)\) and a trinomial \((t^2 - 5t + 5)\). We will use the distributive property to multiply.

Step 2: Distribute the Binomial to Each Term

First, distribute \(t\) from the binomial \((t + 3)\) to each term in the trinomial \((t^2 - 5t + 5)\): 1. \(t \cdot t^2 = t^3\) 2. \(t \cdot (-5t) = -5t^2\) 3. \(t \cdot 5 = 5t\)

Step 3: Continue with Distribution

Next, distribute the constant \(3\) from the binomial \((t + 3)\) to each term in the trinomial: 1. \(3 \cdot t^2 = 3t^2\) 2. \(3 \cdot (-5t) = -15t\) 3. \(3 \cdot 5 = 15\)

Step 4: Combine Like Terms

Now combine all the terms derived from distributing the binomial: \(t^3 - 5t^2 + 5t + 3t^2 - 15t + 15\)Combine the like terms \(-5t^2 + 3t^2\) and \(5t - 15t\): 1. \(-5t^2 + 3t^2 = -2t^2\) 2. \(5t - 15t = -10t\)

Step 5: Write the Final Expression

Combine all the simplified terms to get the final polynomial: \(t^3 - 2t^2 - 10t + 15\)

Key Concepts

Distributive Property

The distributive property is a fundamental principle in mathematics that aids in the simplification of expressions. It helps us multiply a single term by every term inside an expression enclosed in parentheses. In this case, we apply it to a binomial
when multiplying with a trinomial.The distributive property states that for any numbers or terms,
\[a(b + c) = ab + ac.\]It's like laying a carpet (the outside term) over every piece of furniture (inside terms) in a room.
Each piece of furniture gets covered. In this exercise, the binomial
\((t + 3)\) is the carpet, and the trinomial \((t^2 - 5t + 5)\) is the furniture.
We first distribute the \(t\) and then the \(3\) to each term of the trinomial, ensuring
each term is covered by the operation.

Binomial Expansion

Binomial expansion involves multiplying a binomial by another polynomial. Here, we expand by systematically multiplying each term in the binomial,
\((t+3)\), by every term in the trinomial,
\((t^2 - 5t + 5)\). This expansion requires careful application of the distributive property to ensure every combination of terms is considered.Starting with the binomial's first term \( t \):

• \( t \cdot t^2 = t^3 \)
• \( t \cdot (-5t) = -5t^2 \)
• \( t \cdot 5 = 5t \)

Next, expand using the binomial's second term, \(3\):

• \( 3 \cdot t^2 = 3t^2 \)
• \( 3 \cdot (-5t) = -15t \)
• \( 3 \cdot 5 = 15 \)

By covering all possible products, you complete the binomial expansion.

Combine Like Terms

After expanding the expression, you'll arrive at a series of terms. Some of these terms are 'like terms', meaning they have the same variable raised to the same power. Combining them is crucial for simplifying the expression.In the solution provided, the expression was
\(t^3 - 5t^2 + 5t + 3t^2 - 15t + 15\).Notice the like terms:

• \(-5t^2\) and \(3t^2\)
• \(5t\) and \(-15t\)

Combine these terms by adding or subtracting their coefficients:

• \(-5t^2 + 3t^2 = -2t^2\)
• \(5t - 15t = -10t\)

Combining like terms gives you a more simplified, concise expression, necessary for clearer mathematical communication.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. In this exercise, the expression \((t+3)(t^2-5t+5)\) is a perfect example of manipulating algebraic expressions.Variables represent unknown or varying values. Operators indicate mathematical operations such as addition, subtraction, or multiplication.When we multiply these expressions,
we use properties like the distributive property for calculations.Working with algebraic expressions forms the backbone of solving equations. The manipulation of these expressions through expansion or simplification teaches critical thinking and problem-solving. Understanding these concepts helps in approaching more complex algebraic problems with confidence.