Problem 22 - College Algebra

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

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

Step 1: Use the Distributive Property

To multiply two binomials, use the distributive property, often referred to as the FOIL method for binomials. Multiply each term in the first binomial by each term in the second binomial. Layout the multiplication first: \[(4t^2 + 7t)(-3t^2 + 4)\]. Expand it as follows: \[4t^2 \cdot (-3t^2),\ 4t^2 \cdot 4,\ 7t \cdot (-3t^2),\ 7t \cdot 4\].

Step 2: Perform Each Multiplication

Calculate each of the terms from the multiplication: - \(4t^2 \cdot (-3t^2) = -12t^4\)- \(4t^2 \cdot 4 = 16t^2\)- \(7t \cdot (-3t^2) = -21t^3\)- \(7t \cdot 4 = 28t\)

Step 3: Combine Like Terms

Now combine all the calculated terms from the previous step: \(-12t^4, -21t^3, 16t^2, 28t\). Since these terms all have different exponents, they cannot be combined further. The product of the original binomials is: \(-12t^4 - 21t^3 + 16t^2 + 28t\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by each term inside a parenthesis. In the context of binomial multiplication, this property shows its full potential. When multiplying two binomials, such as \((a + b)(c + d)\), you use the distributive property to ensure each term in the first binomial is multiplied by each term in the second binomial. In this way, the expression expands to:

• First, multiply \(a\) with \(c\) and \(d\).
• Then, multiply \(b\) with \(c\) and \(d\).

Resulting in an expression like \(ac + ad + bc + bd\). In our original problem \((4t^2 + 7t)(-3t^2 + 4)\), we apply the distributive property by multiplying each term as demonstrated:

• \(4t^2 \cdot (-3t^2) \)
• \(4t^2 \cdot 4 \)
• \(7t \cdot (-3t^2) \)
• \(7t \cdot 4 \)

This ensures that every combination is accounted for, resulting in a more straightforward expansion and solution.

FOIL Method

The FOIL method is a handy trick used to simplify the multiplication of two binomials. "FOIL" stands for First, Outer, Inner, and Last, which refers to the order in which you multiply the terms:

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

By using the FOIL method on the expression \((4t^2 + 7t)(-3t^2 + 4)\), you perform the following steps:

• **First:** \(4t^2 \cdot (-3t^2) = -12t^4\)
• **Outer:** \(4t^2 \cdot 4 = 16t^2\)
• **Inner:** \(7t \cdot (-3t^2) = -21t^3\)
• **Last:** \(7t \cdot 4 = 28t\)

The FOIL method is essentially the distributive property applied in a specific sequence. It's a quick and efficient strategy for multiplying binomials.

Polynomial Arithmetic

Polynomial arithmetic involves operations with polynomials, such as addition, subtraction, multiplication, and sometimes division. Understanding how to handle these operations is crucial for mastering algebra.When multiplying polynomials, such as binomials in our exercise, we rely heavily on multiplication rules like those seen in the distributive property or FOIL method. After performing multiplication, the next step in polynomial arithmetic is often combining like terms. Like terms in a polynomial are terms that have the same variable raised to the same power. In this context:

• The result of our binomial multiplication gives us a polynomial \(-12t^4 - 21t^3 + 16t^2 + 28t\).
• These terms do not have any like terms that can be combined further since each term has a different power of \(t\).

This means the expression is already in its simplest polynomial form, ready for any further algebraic manipulations or usage in equations. In polynomial arithmetic, mastering these basic operations builds a foundation for more complex algebraic concepts.