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Chapter 5: Problem 112

If \(R(x)=x+5, Q(x)=x^{2}-2,\) and \(P(x)=5 x,\) find the following. $$ P(x) \cdot Q(x) $$

Short Answer

Expert verified
The result of \( P(x) \cdot Q(x) \) is \( 5x^3 - 10x \).

Step by step solution

01

Write the functions

We have three given functions: 1. \( R(x) = x + 5 \)2. \( Q(x) = x^2 - 2 \)3. \( P(x) = 5x \)We need to find the expression for \( P(x) \cdot Q(x) \).
02

Multiply the functions

We want to find \( P(x) \cdot Q(x) \). This means we will multiply the expressions for \( P(x) \) and \( Q(x) \):\[ P(x) \cdot Q(x) = (5x) \cdot (x^2 - 2) \]
03

Distribute the multiplication

To simplify, distribute \(5x\) into the parenthesis:\[ 5x imes (x^2) + 5x imes (-2) \]Which simplifies to:\[ 5x^3 - 10x \]
04

Simplify the expression

The expression from Step 3 is already simplified. Therefore, the result of \( P(x) \cdot Q(x) \) is:\[ 5x^3 - 10x \].

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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 that consist of variables and coefficients, constructed using addition, subtraction, and multiplication. They may also involve variables raised to whole number exponents. These functions are fundamental in algebra and are often used to solve various types of equations.

A polynomial function can have one or more terms, such as constant terms, linear terms, quadratic terms, and so forth. This variety gives them flexibility in modeling different scenarios.
  • Constant polynomials like 5 are terms that don't change.
  • Linear polynomials might look like \(x + 5\), involving the first power of \(x\).
  • Quadratic polynomials, such as \(x^2 - 2\), involve the second power of \(x\).
Understanding polynomial functions is the first step to mastering operations involving them.
Multiplication of Polynomials
Multiplying polynomials involves distributing each term of the first polynomial across every term of the second polynomial. This method is commonly known as the distributive property.

Let鈥檚 break down the multiplication in our exercise example:
  • Consider two polynomials: \(P(x) = 5x\) and \(Q(x) = x^2 - 2\).
  • The task is to multiply these: \( P(x) \cdot Q(x) = (5x)(x^2 - 2)\).
  • Use the distributive property to multiply \(5x\) by each term in \(Q(x)\).
The result is that each term in one polynomial is scaled by each term in the other, ensuring all interactions between terms are captured. This results in new polynomial terms where exponents and coefficients are combined. In our example, this gives us \(5x^3 - 10x\).
Algebraic Expressions
Algebraic expressions, such as the ones we have used in this exercise, form the core of algebraic operations. These expressions consist of numbers, variables, and arithmetic operations. The goal when working with algebraic expressions is often to simplify them to understand better or solve problems.

Here are some important points about algebraic expressions:
  • They include terms like \(5x\) and \(x^2\), which are separated by addition or subtraction.
  • Simplification often involves combining like terms, which have the same variable raised to the same power.
  • Operations on algebraic expressions, like our example of polynomial multiplication, help to solve equations and analyze functions.
Simplifying expressions makes complex problems more manageable and reveals underlying patterns or solutions. Thus, developing proficiency in manipulating these expressions is key to succeeding in algebra and advanced mathematics.

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Most popular questions from this chapter

Factor. Assume that variables used as exponents represent positive integers. $$ 3 x^{2 n}-8 x^{n}+4 $$

Factor each polynomial completely. See Examples 1 through 12. $$ x^{2}+6 x y+5 y^{2} $$

Recall that a graphing calculator may be used to check addition, subtraction, and multiplication of polynomials. In the same manner, a graphing calculator may be used to check factoring of polynomials in one variable. For example, to see that $$ 2 x^{3}-9 x^{2}-5 x=x(2 x+1)(x-5) $$ graph \(\mathrm{Y}_{1}=2 x^{3}-9 x^{2}-5 x\) and \(\mathrm{Y}_{2}=x(2 x+1)(x-5) .\) Then trace along both graphs to see that they coincide. Factor the following and use this method to check your results. $$ x^{3}+6 x^{2}+8 x $$

Factor. Assume that variables used as exponents represent positive integers. $$ x^{2 n}+7 x^{n}-18 $$

Multiply. See Section 5.4. $$ (x-4)(x+4) $$
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