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Chapter 4: Problem 51

In Problems 50 and 51 , if \(f(x)=x^{2}+2 x-7\) and \(g(x)=3 x-4\) find: 51\. \((f \cdot g)(x)\)

Short Answer

Expert verified
(f 路 g)(x) = 3x^3 + 2x^2 - 29x + 28

Step by step solution

01

Write Down the Functions

First, note the given functions: \[ f(x) = x^2 + 2x - 7 \] and \[ g(x) = 3x - 4 \]
02

Define the Product of the Functions

The product of the functions \(f \cdot g\)(x) is defined as: \[ (f \cdot g)(x) = f(x) \cdot g(x) \]
03

Multiply the Functions

To find \(f \cdot g\)(x), multiply \( f(x) \) by \( g(x) \): \[ (f \cdot g)(x) = (x^2 + 2x - 7) \cdot (3x - 4) \]Distribute each term in \((3x - 4)\) by each term in \((x^2 + 2x - 7)\).
04

Distribute and Combine Like Terms

Perform the multiplication: \[ (x^2 + 2x - 7) \cdot (3x) = 3x^3 + 6x^2 - 21x \] and \[ (x^2 + 2x - 7) \cdot (-4) = -4x^2 - 8x + 28 \] Combine these results: \[ (f \cdot g)(x) = 3x^3 + 6x^2 - 21x - 4x^2 - 8x + 28 \] Combine like terms: \[ (f \cdot g)(x) = 3x^3 + 2x^2 - 29x + 28 \]

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Key Concepts

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

Polynomial Functions
Polynomial functions are expressions involving variables raised to whole number powers and coefficients. They often appear in the form:
  • anxn + an-1xn-1 + ... + a1x + a0
where each ai is a coefficient and x is the variable. In the given problem, we have two polynomial functions: f(x) = x2 + 2x - 7 and g(x) = 3x - 4.Notice that f(x) is a second-degree polynomial (highest power is x2) and g(x) is a first-degree polynomial (highest power is x).These forms are important because they dictate how the functions behave and how we can manipulate them, like in the multiplication we are doing here.
Distributive Property
The distributive property is essential when multiplying polynomials. It states that c(a + b) = ca + cb.This means we need to multiply each term in the first polynomial by every term in the second polynomial. Applying this to our problem, we multiply f(x) = x2 + 2x - 7 by g(x) = 3x - 4in stages:
  • Multiply each term of (3x - 4) by each term of (x2 + 2x - 7).
This looks like:
  • (x2 + 2x - 7) * 3x = 3x3 + 6x2 - 21x
  • (x2 + 2x - 7) * (-4) = -4x2 - 8x + 28
This step-by-step multiplication follows the distributive property, ensuring all terms are correctly multiplied.
Combining Like Terms
After applying the distributive property, we get a series of terms that need to be combined. Combining like terms means adding or subtracting terms with the same power of x. In our schedule, the results from (x2 + 2x - 7) * 3x = 3x3 + 6x2 - 21xand (x2 + 2x - 7) * (-4) = -4x2 - 8x + 28must be added together: Combining like terms:
  • The x3 terms: 3x3
  • The x2 terms: 6x2 - 4x2 = 2x2
  • The x terms: -21x - 8x = -29x
  • The constant terms: 28
So,Combining all, we get: 3x3 + 2x2 - 29x + 28.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. In our exercise, we work with two algebraic expressions: f(x) = x2 + 2x - 7 and g(x) = 3x - 4.Multiplying these expressions as we did involves understanding how to handle the variables, coefficients, and terms. Understanding the structure of algebraic expressions is key to solving multiplication problems. Each step, such as applying the distributive property and combining like terms, relies on recognizing how algebraic structures interact. Working through exercises step by step improves proficiency in manipulating these expressions accurately.

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Most popular questions from this chapter

Why is the graph of a quadratic function concave up if \(a>0\) and concave down if \(a<0 ?\)

The simplest cost function \(C\) is a linear cost function, \(C(x)=m x+b,\) where the \(y\) -intercept \(b\) represents the fixed costs of operating a business and the slope \(m\) represents the cost of each item produced. Suppose that a small bicycle manufacturer has daily fixed costs of \(\$ 1800,\) and each bicycle costs \(\$ 90\) to manufacture. (a) Write a linear model that expresses the cost \(C\) of manufacturing \(x\) bicycles in a day. (b) Graph the model. (c) What is the cost of manufacturing 14 bicycles in a day? (d) How many bicycles could be manufactured for \(\$ 3780 ?\)

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