Question

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

Short answer

The product of the binomials \((4-5x)(1-2x)(3x+2)\) is \(30x^3 - 19x^2 - 18x + 8\).

Step-by-step solution

Multiplication of the First Two Binomials

The first step is to multiply the first two binomials: \((4-5x)(1-2x)\). Apply the distributive property also known as FOIL (which stands for First, Outer, Inner, Last). The product is \((4- 8x - 5x +10x^2)\) which simplifies to \((10x^2 - 13x + 4)\)

Multiplication of the result and the last binomial

The second step is to multiply the result of the first product by the last binomial. So, now we multiply \((10x^2 - 13x + 4)\) with \((3x+2)\). Again applying the distributive property, the product is \((30x^3 - 39x^2 + 8x + 20x^2 - 26x + 8)\), which simplifies to \((30x^3 - 19x^2 - 18x + 8)\)

Final Result

The final result is the simplified form of the product, which is \(30x^3 - 19x^2 - 18x + 8\)

Key concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions and solve equations. It's especially useful when multiplying polynomials, like binomials, by breaking down complex expressions into more manageable parts. In our exercise, to find the product of the binomials

• This property lets us distribute each term of one binomial to every term of another.
• For instance, when multiplying a term from the first parentheses multiplies every term in the second parentheses.
• So, when we took (4-5x) and (1-2x), each term from the first binomial was multiplied by each term from the second binomial.

This process, often called FOILing when applied to two binomials, breaks into several products — First, Outer, Inner, and Last terms. The final step is combining these products to obtain a single expression. With practice, mastering the distributive property helps simplify the process of tackling multiple polynomials.

Binomials

A binomial is a polynomial with two terms separated by a plus (+) or minus (-) sign. This exercise involves finding the product of three binomials, which are expressions consisting of two terms each. In our case:

• The binomials given are (4-5x), (1-2x), and (3x+2).
• Each of these binomials represents a foundational polynomial type and provides the structure needed to apply multiplication across them.

When working with binomials, it is important to perform operations methodically. We first multiplied (4-5x) by (1-2x) to gain an intermediate result: (10x^2 - 13x + 4). Then, this intermediate result was further multiplied by the third binomial, (3x+2), to eventually reach the final simplified product. Breaking down this way ensures all parts of the binomials are multiplied correctly, which is crucial for achieving the correct result.

Simplifying Expressions

Simplifying expressions is the process of making a mathematical expression as concise and straightforward as possible. It often involves collecting like terms, which are terms that share the same variables and powers, and performing operations to reduce the expression into a simpler form.

• Throughout the given problem, after applying the distributive property, you'll need to simplify multiple times.
• For example, after multiplying the first two binomials, we simplify (4 - 8x - 5x + 10x^2) to (10x^2 - 13x + 4) by combining like terms.
• Subsequently, once all binomials have been multiplied, further simplification is required, leading to the final result: (30x^3 - 19x^2 - 18x + 8).

This final step of simplification is critical for clarity and to ensure that the polynomial is expressed in its simplest form, making it easier to interpret and analyze. The ability to simplify expressions effectively is a valuable skill in algebra and other mathematical disciplines.