In Exercises 15–58, find each product. (2x−3)(x2−3x+5)

Ch. P - Fundamental Concepts of Algebra

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Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

Problem 17

Chapter 1, Problem 17

In Exercises 15–58, find each product. (2x−3)(x²−3x+5)

Verified step by step guidance

Step 1: Recognize that this is a multiplication of two polynomials, specifically a binomial (2x - 3) and a trinomial (x^2 - 3x + 5). To solve, use the distributive property (also known as the FOIL method for binomials).

Step 2: Distribute the first term of the binomial, 2x, to each term in the trinomial. This gives: (2x * x^2) + (2x * -3x) + (2x * 5).

Step 3: Simplify the terms from Step 2. This results in: 2x^3 - 6x^2 + 10x.

Step 4: Distribute the second term of the binomial, -3, to each term in the trinomial. This gives: (-3 * x^2) + (-3 * -3x) + (-3 * 5).

Step 5: Simplify the terms from Step 4 and combine them with the terms from Step 3. This results in: 2x^3 - 6x^2 + 10x - 3x^2 + 9x - 15. Combine like terms to simplify further.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Multiplication

Polynomial multiplication involves distributing each term of one polynomial to every term of another polynomial. This process is often referred to as the distributive property, where you multiply each term in the first polynomial by each term in the second. For example, in the expression (2x−3)(x^2−3x+5), you would multiply 2x by each term in the second polynomial and then do the same for -3.

Combining Like Terms

After multiplying polynomials, the next step is to combine like terms, which are terms that have the same variable raised to the same power. This simplification process helps in reducing the expression to its simplest form. For instance, if the multiplication yields terms like 6x^2 and -3x^2, you would combine them to get 3x^2.

Standard Form of a Polynomial

The standard form of a polynomial is when the terms are arranged in descending order of their degrees. This means that the term with the highest exponent comes first, followed by the next highest, and so on. For example, a polynomial like 3x^2 + 2x - 5 is in standard form, which is important for clarity and further operations such as addition or comparison with other polynomials.

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