Question

s 27–32, fi nd the product of the binomials \((3 x-4)(5-2 x)(4 x+1)\)

Short answer

The product of the binomials is \( -24x^3 + 86x^2 - 57x - 20 \)

Step-by-step solution

Multiply the first two binomials

Using the distributive or FOIL property, \( (3x - 4)(5 - 2x) = 15x - 6x^2 -20 + 8x = -6x^2 + 23x - 20 \)

Multiply resulting first product by the third binomial

Again using the FOIL property, \( (-6x^2 + 23x - 20)(4x + 1) = -24x^3 +92x^2 - 80x -6x^2 +23x - 20 = -24x^3 + 86x^2 - 57x - 20 \)

Solution

Therefore, the product of \( (3x - 4)(5 - 2x)(4x + 1) = -24x^3 + 86x^2 - 57x - 20 \)

Key concepts

Understanding Binomials

In algebra, a binomial is simply a type of polynomial that contains exactly two terms. These terms are usually joined by a plus or minus sign. Binomials are fundamental in algebra because they form the basis for more complex expressions. For example, in the expression \(3x - 4\), '3x' and '4' are the two terms, and they are connected by a subtraction sign.

Binomials can be added, subtracted, multiplied, and divided just like other expressions. However, when multiplying binomials, special methods such as the FOIL method are often used to simplify the process. Understanding how to work with binomials is a critical skill in algebra that helps in solving more complicated equations.

Distributive Property in Multiplication

The distributive property is a key concept in algebra that is crucial when multiplying polynomials, including binomials. It states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. In a mathematical context, the distributive property is expressed as \(a(b + c) = ab + ac\).

This property is particularly useful when dealing with binomials and other polynomials as it allows us to break down complex multiplication into simpler, more manageable steps. For example, when multiplying the binomials \((3x - 4)(5 - 2x)\), the distributive property lets us multiply each term in the first binomial by each term in the second binomial.

• First, multiply '3x' with every term in \(5 - 2x\).
• Then, multiply '-4' with every term in \(5 - 2x\).
• Finally, add all the individual products together to get the final expression.

Mastering the FOIL Method

The FOIL method is a specialized application of the distributive property used to multiply two binomials. FOIL is an acronym that stands for First, Outer, Inner, and Last. It represents the four steps used to multiply the terms in each binomial.

To apply the FOIL method, you would:

• Multiply the First terms of each binomial.
• Multiply the Outer terms of each binomial.
• Multiply the Inner terms of each binomial.
• Multiply the Last terms of each binomial.

Once you've multiplied these pairs, add all the products together to get the final result. For the provided example, when multiplying \((3x - 4)(5 - 2x)\), following the FOIL steps yields the products: \(15x\), \(-6x^2\), \(-20\), and \(8x\). Combining these products simplifies the expression to \(-6x^2 + 23x - 20\).

The FOIL method helps streamline the multiplication process and is a quick tool for confirming your work when multiplying binomials.

Exploring Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operation signs that represent a particular value or set of values. Unlike a simple numerical equation, algebraic expressions can include variables like \(x\) and \(y\), which can represent unknown numbers. These forms of expressions allow mathematicians to model real-world situations and solve various problems.

In the context of multiplying polynomials, algebraic expressions become particularly useful as they need to account for each variable and constant. For instance, the expression \((-6x^2 + 23x - 20)(4x + 1)\) includes variables raised to different powers and demonstrates how complex algebraic expressions can become.

• These expressions often require careful use of algebraic rules, such as the distributive property and FOIL method, to simplify and find solutions.
• Algebraic expressions need to be simplified methodically to ensure no computation errors creep in.

The ability to understand and manipulate algebraic expressions is essential for solving equations of all types and understanding more advanced mathematical concepts.