/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Multiply the polynomials. $$(6... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 30

Multiply the polynomials. $$(6 x+5)(3 x-1)$$

Short Answer

Expert verified
18x^2 + 9x - 5

Step by step solution

01

- Distribute Each Term in the First Polynomial

Multiply each term in the first polynomial, (6x + 5), by each term in the second polynomial, (3x - 1). This gives us: \[ 6x \times 3x, 6x \times (-1), 5 \times 3x, 5 \times (-1) \]
02

- Perform the Multiplications

Calculate each of the products: \[ 6x \times 3x = 18x^2 \ 6x \times (-1) = -6x \ 5 \times 3x = 15x \ 5 \times (-1) = -5 \]
03

- Combine Like Terms

Now, add the results of the multiplications together: \[ 18x^2 - 6x + 15x - 5 \] Combine the like terms (-6x and 15x):\[ 18x^2 + 9x - 5 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

distributive property
To understand polynomial multiplication, we first need to grasp the princip le of the distributive property. The distributive property states that for any three numbers, a, b, and c: a(b + c) = ab + ac . In the context of polynomials, each term in the first polynomial has to be multiplied by each term in the second polynomial. Let's break it down using our example: (6x + 5)(3x - 1). Here, you'll multiply each term in the first polynomial (6x and 5) by each term in the second polynomial (3x and -1). This step ensures that every possible product is accounted for. Remember to keep track of the signs (positive and negative) while performing these multiplications.
combining like terms
After distributing all the terms and performing the multiplications, we get several new terms. Now, it's time to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, 6x and -2x are like terms because they both contain 'x' to the first power. To combine like terms, add or subtract the coefficients (the numerical part) of these terms. In our example, after multiplication, we have: 18x^2 - 6x + 15x - 5 . Notice that -6x and 15x are like terms because they both have the variable 'x'. So, we combine them by adding their coefficients: -6 + 15 = 9.
polynomials
To understand polynomial multiplication, it's important to first understand what polynomials are. A polynomial is an algebraic expression made up of terms, where each term consists of a coefficient (numeric part) and a variable (like x) raised to a non-negative integer exponent. For example, 3x^2 + 2x - 5 is a polynomial with three terms. Each term's degree is defined by its exponent. The degree of a polynomial is the highest degree of its terms. For example, the degree of 3x^2 + 2x - 5 is 2 because that's the highest exponent.

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Most popular questions from this chapter

Perform the indicated operations. $$\left(\frac{7}{8} a-4 b-\frac{1}{2} c\right)-\left(-\frac{1}{4} a+\frac{1}{2} b+\frac{3}{8} c\right)$$

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Simplify the expression. $$\left(2 x y^{4}\right)^{3}\left(4 x^{2} y^{3}\right)$$

The cost (in dollars) for a cab ride is given by the polynomial \(2+1.25 x+\) \(0.25 y .\) In this context, \(x\) is the number of miles driven and \(y\) is the time in minutes for the ride. a. Evaluate the polynomial for \(x=10\) and \(y=20 .\) Interpret the answer in the context of this problem. b. Evaluate the polynomial for \(x=22\) and \(y=40 .\) Interpret the answer in the context of this problem.
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