Problem 29 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ (3 t-2)(7 t-5) $$

Step-by-Step Solution

Answer

The simplified expression is \(21t^2 - 29t + 10\).

Step 1: Apply the Distributive Property

To simplify \((3t-2)(7t-5)\), apply the distributive property (also known as the FOIL method for binomials). Each term in the first binomial \((3t - 2)\) must be multiplied by each term in the second binomial \((7t - 5)\).

Step 2: Multiply the First Terms

First, multiply the first terms of each binomial: \(3t imes 7t = 21t^2\). This will be the first term of our product.

Step 3: Multiply the Outer Terms

Next, multiply the outer terms: \(3t imes -5 = -15t\). This provides the next term of the product.

Step 4: Multiply the Inner Terms

Now, multiply the inner terms: \(-2 imes 7t = -14t\). This gives us another term for the product.

Step 5: Multiply the Last Terms

Finally, multiply the last terms: \(-2 imes -5 = 10\). This is the final term of the product.

Step 6: Combine Like Terms

Combine all the terms from the previous steps: \(21t^2\), \(-15t\), \(-14t\), and \(+10\). Adding the like terms \(-15t\) and \(-14t\) gives \(-29t\). Therefore, the expression simplifies to \(21t^2 - 29t + 10\).

Key Concepts

Distributive Property

The distributive property is a fundamental building block in algebra, allowing us to simplify complex expressions. It applies to operations like multiplication across addition or subtraction, essentially distributing the multiplier to each of the terms within the brackets. In the expression

• \((3t-2)(7t-5)\),

each element of the first binomial is multiplied by each element in the second one. This method ensures that we do not miss any interactions between the terms, as every single component is accounted for and multiplied accordingly. The property is an essential tool used to break down expressions into more manageable parts, making calculations simpler and reducing the risk of errors.

This principle is the basis for the FOIL method used in binomial multiplication, which we'll explore next.

Binomial Multiplication

Binomial multiplication involves multiplying two binomial expressions, like in the exercise

• \((3t-2)(7t-5)\).

A commonly used technique for this is the FOIL method, an acronym for First, Outer, Inner, Last, which helps in the systematic multiplication of each term. - **First terms**: Multiply the first terms from each binomial. In our example, \(3t \times 7t = 21t^2\).- **Outer terms**: Multiply the outer terms, which are \(3t\) and \(-5\). Here, we have \(3t \times -5 = -15t\).- **Inner terms**: Multiply the inner terms, \(-2\) and \(7t\), giving us \(-2 \times 7t = -14t\).- **Last terms**: Finally, multiply the last terms from each binomial. For us, \(-2 \times -5 = 10\).This method ensures a thorough multiplication process, covering all possible interactions between the terms in the binomials.

Polynomial Simplification

Polynomial simplification is the process of combining like terms to create a cleaner, more concise expression. After multiplying the binomials

• \((3t-2)(7t-5)\),

we end up with several terms: \(21t^2, -15t, -14t,\) and \(10\). The simplification process involves grouping and combining these like terms to reduce the expression to its simplest form.- The terms \(-15t\) and \(-14t\) share the same variable component \(t\), so they can be summed to get \(-29t\).- The constant term \(10\) remains unchanged, as it does not have any other similar constant to combine with.Therefore, the final simplified expression becomes \(21t^2 - 29t + 10\). Simplifying polynomials in this way helps to clearly present their value and behavior, making it easier to solve algebraic problems.