Problem 42 - Algebra and Trigonometry

Question

Multiply: \((2 x-7)\left(3 x^{2}-5 x+4\right)\)

Step-by-Step Solution

Answer

The product is \(6x^3 - 31x^2 + 43x - 28\).

Step 1 - Distribute the first term

Multiply the first term in the first binomial, which is \(2x\), by each term in the second polynomial \(3x^2 - 5x + 4\).

Step 2 - Calculate individual products

Calculate the products for each multiplication step: 1. \(2x \times 3x^2 = 6x^3\) 2. \(2x \times (-5x) = -10x^2\) 3. \(2x \times 4 = 8x\)

Step 3 - Distribute the second term

Multiply the second term in the first binomial, which is \(-7\), by each term in the second polynomial \(3x^2 - 5x + 4\).

Step 4 - Calculate individual products

Calculate the products for each multiplication step: 1. \(-7 \times 3x^2 = -21x^2\) 2. \(-7 \times (-5x) = 35x\) 3. \(-7 \times 4 = -28\)

Step 5 - Combine like terms

Add the results from Steps 2 and 4 together: \[6x^3 - 10x^2 + 8x - 21x^2 + 35x - 28\]Combine like terms: \[6x^3 - 31x^2 + 43x - 28\]

Key Concepts

binomial distribution

Binomial distribution refers to distributing each term in a binomial expression across the terms of another polynomial. In this exercise, we start by distributing each term of the binomial \(2x-7\) across the polynomial \(3x^2-5x+4\).

We first distribute \(2x\), multiplying it with each term of the second polynomial:

• \(2x \times 3x^2 = 6x^3\)
• \(2x \times -5x = -10x^2\)
• \(2x \times 4 = 8x\)

Then, we distribute the second term \(-7\) similarly:

• \(-7 \times 3x^2 = -21x^2\)
• \(-7 \times -5x = 35x\)
• \(-7 \times 4 = -28\)

This method ensures that all interactions between terms are considered.

combining like terms

Combining like terms means adding or subtracting coefficients of terms that have the same variables raised to the same power. In the final step of our exercise, we combine the results from distributing \(2x\) and \(-7\):

We start with \[6x^3 - 10x^2 + 8x - 21x^2 + 35x - 28\],
and then group and combine the like terms:

• \[6x^3\] (no like term to combine)
• \[-10x^2 - 21x^2 = -31x^2\] (combine the \(-x^2\) terms)
• \[8x + 35x = 43x\] (combine the \(+x\) terms)
• \[-28\] (no like term to combine)

So, we get \[6x^3 - 31x^2 + 43x - 28.\]

distributive property

The distributive property of multiplication lets us multiply a single term across terms inside a parenthesis. It's an essential property used in polynomial multiplication.

In our exercise, we use the distributive property twice—once for each term in the binomial \(2x-7\).

First, we distribute \(2x\) across \(3x^2-5x+4\) as follows:

• \(2x \times 3x^2\)
• \(2x \times -5x\)
• \(2x \times 4\)

Next, we distribute \(-7\):

• \(-7 \times 3x^2\)
• \(-7 \times -5x\)
• \(-7 \times 4\)

This property ensures that we account for all product terms.

polynomial arithmetic

Polynomial arithmetic involves operations like addition, subtraction, and multiplication of polynomials. In this exercise, we focus on multiplication.

When multiplying polynomials such as a binomial and a trinomial, we distribute each term in the binomial across each term in the trinomial. Here's a reminder of the steps:

• Distribute the first term of the binomial.
• Calculate the individual products.
• Distribute the second term of the binomial.
• Calculate the individual products.
• Combine like terms to simplify.

Polynomials can have multiple terms, so being systematic helps ensure no term is missed. Mastering polynomial arithmetic is essential for more advanced algebra topics.