Problem 22 - College Algebra with Corequisite Support

Question

For the following exercises, find the product. \(\left(4 t^{2}+7 t\right)\left(-3 t^{2}+4\right)\)

Step-by-Step Solution

Answer

The product is \(-12t^4 - 21t^3 + 16t^2 + 28t\).

Step 1: Apply the Distributive Property

To find the product of the two binomials \((4t^2 + 7t)(-3t^2 + 4)\), use the distributive property (also known as the FOIL method for binomials). This involves multiplying each term in the first binomial by each term in the second binomial.

Step 2: Multiply Each Pair of Terms

Multiply each term in the first binomial \(4t^2 + 7t\) with each term in the second binomial \(-3t^2 + 4\). This results in four products:1. \((4t^2) \times (-3t^2) = -12t^4\)2. \((4t^2) \times 4 = 16t^2\)3. \((7t) \times (-3t^2) = -21t^3\)4. \((7t) \times 4 = 28t\).

Step 3: Combine All Terms

Combine the four products found in Step 2 to form a single expression:\(-12t^4 - 21t^3 + 16t^2 + 28t\).

Step 4: Simplify the Expression

Since there are no like terms to combine in the expression \(-12t^4 - 21t^3 + 16t^2 + 28t\), this is the simplified product of the original binomials.

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra that helps in simplifying expressions and is vital for multiplying polynomials, such as binomials. It states that the multiplication of a single term by a sum of two or more terms is equal to the sum of the individual products of the single term and each of the terms within the parentheses. You can think of it as distributing the multiplication across each term inside the parentheses.

For example, when multiplying two binomials like \((4t^2 + 7t) \) and \(( -3t^2 + 4)\), the distributive property ensures that each term in the first binomial multiplies each term in the second binomial. This results in multiple products which are then summed together.

• Multiply \((4t^2)\) with \((-3t^2)\) and \(4\).
• Multiply \((7t)\) with \((-3t^2)\) and \(4\).

Understanding this property makes tackling polynomial expressions much simpler and more organized.

Polynomial Expressions

Polynomial Expressions are combinations of variables and coefficients formed under mathematical operations such as addition, subtraction, and multiplication. They can vary in complexity, ranging from simple one-term monomials to complex multi-term polynomials. Each term in a polynomial is composed of a product of constants and non-negative integer powers of variables.

In the exercise provided, two polynomials are being multiplied. Specifically, each polynomial, or binomial in this case, consists of two terms:

• One is \(4t^2 + 7t\)
• The other is \(-3t^2 + 4\)

Combining these through multiplication requires the application of both the Distributive Property and other algebraic techniques. Recognizing polynomial structures is crucial, as it determines the method you will use for manipulation and simplification.

Algebraic Simplification

Algebraic Simplification refers to the process of reducing mathematical expressions to their simplest form. This involves combining like terms and using mathematical properties, such as the Distributive Property, to make expressions more concise.

In this particular exercise, simplification becomes important after multiplying the terms to form the expression:

• \(-12t^4\)
• \(-21t^3\)
• \(16t^2\)
• \(28t\)

Since each resulting term is unique and contains a different power of the variable \(t\), none are like terms. Therefore, no further simplification by combining terms is possible. The full expression, \(-12t^4 - 21t^3 + 16t^2 + 28t\), is already in its simplest form.

FOIL Method

The FOIL Method is a technique used specifically for multiplying two binomials. It provides a systematic way to apply the Distributive Property, focusing on the multiplication of specific pairs of terms. FOIL stands for "First, Outer, Inner, Last," referring to the position of terms in the two binomials.

When applying the FOIL Method to \(4t^2 + 7t\) and \(-3t^2 + 4\), the process is as follows:

• First: Multiply the first terms: \(4t^2 \ imes -3t^2 = -12t^4\).
• Outer: Multiply the outer terms: \(4t^2 \ imes 4 = 16t^2\).
• Inner: Multiply the inner terms: \(7t \ imes -3t^2 = -21t^3\).
• Last: Multiply the last terms: \(7t \ imes 4 = 28t\).

While the FOIL Method is specifically useful for binomials, the underlying principles aid in understanding larger polynomial multiplications. Remembering FOIL provides a clear roadmap for binomial multiplication.