Question

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

Short answer

The product of the binomials \( (x - 2) \), \( (3x + 1) \), and \( (4x - 3) \) is \( 12x^3 - 29x^2 + 7x + 6 \).

Step-by-step solution

Multiply the first two binomials

First, multiply the binomials \( (x - 2) \) and \( (3x + 1) \) using the distributive property. Distribute each term in the first binomial by each term in the second: \( x*3x + x*1 - 2*3x - 2*1 = 3x^2 + x - 6x - 2 \). Simplify the result to obtain the polynomial \(3x^2 - 5x -2 \).

Multiply the result by the third binomial

Next, multiply the resulting polynomial \( 3x^2 - 5x - 2 \) by the remaining binomial \( (4x - 3) \) using the distributive property. Distribute each term in the binomial by each term in the polynomial: \( 4x*3x^2 - 4x*5x + 4x*-2 - 3*3x^2 + 3*5x + 3*2 \). Simplify the result to obtain the trinomial \( 12x^3 - 20x^2 -8x - 9x^2 + 15x + 6 \).

Simplify the resulting trinomial

Finally, by combining like terms \( -20x^2 -9x^2 \) to get \( -29x^2 + 15x + 6 \), resulting in the final polynomial \( 12x^3 - 29x^2 + 7x + 6 \).

Key concepts

Distributive Property

The distributive property is a critical concept in algebra that allows us to multiply a single term by a group of terms inside a parenthesis. When multiplying binomials, we use the distributive property to ensure that each term in the first binomial multiplies by each term in the second binomial. For example, looking at the exercise ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline Consider the multiplication of the binomials ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline , we distribute the terms as follows: ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline . This process is also known as 'FOIL' (First, Outside, Inside, Last), which refers to multiplying the first, outside, inside, and last terms in each binomial. Applying the distributive property step by step helps to ensure accuracy and prevent errors in multiplication.

Polynomial Simplification

Polynomial simplification involves reducing expressions into their simplest form. While working with polynomials, after applying the distributive property, we may end up with a longer expression that could be simplified. Simplifying a polynomial entails combining like terms and arranging them in descending order of their degrees. Like terms are terms whose variables and their corresponding exponents are identical. ewline ewline When simplifying the product of binomials, we look for terms with the same degree. For instance, after multiplying the first two binomials in our example, we get terms like ewline ewline ewline . We then combine terms with the same degree—in this case, ewline ewline ewline —and simplify them to get ewline ewline ewline . In essence, simplification reduces the polynomial to its most concise and simplified state, which can be more easily understood and used in further mathematical processes.

Combining Like Terms

Combining like terms is the process of merging terms that share the same variable factors and exponents into a single term. This is crucial to simplify expressions, including the result of polynomial multiplication. Once we have distributed and multiplied the terms in the binomials, we can often find terms that can be combined because they are like terms. ewline ewline For instance, after multiplying the binomials in our exercise, we obtain various terms such as ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline and ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline . To combine like terms, we add or subtract their coefficients: in our example, ewline ewline ewline combined to form ewline ewline ewline . This step makes the final result much neater, resulting in the simplified polynomial ewline ewline ewline . Consequently, understanding and applying the concept of combining like terms is indispensable for students dealing with the simplification of polynomials.