Problem 103 Let $P(x)=3 x-5 .$ Find each v... [FREE SOLUTION]

91影视

Beginning Algebra: A Guided Approach

Rosemary Karr, Marilyn Massey, R. David Gustafson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 4: Problem 103

Let $P(x)=3 x-5 .$ Find each value. $$P(x+h)+P(x)$$

Short Answer

Expert verified

The value of the expression $P(x+h) + P(x)$ for the polynomial $P(x)=3x-5$ is $6x + 3h - 10$.

Step by step solution

Substitute $x+h$ into $P(x)$

To find the value of $P(x + h)$, replace every $x$ in $P(x) = 3x - 5$ with $(x + h)$. This gives $P(x + h) = 3(x + h) - 5 = 3x + 3h - 5$.

Determine $P(x)$

The polynomial $P(x)$ is already given by $3x - 5$. There's no need for further substitutions or simplifications.

Add $P(x + h)$ and $P(x)$

We have $P(x + h) = 3x + 3h - 5$ and $P(x) = 3x - 5$. Adding these two together as per the original problem gives: $P(x + h) + P(x) = (3x + 3h - 5) + (3x - 5) = 6x + 3h - 10$.

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Key Concepts

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

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operators (like addition, multiplication, etc.). In the context of this exercise, expressions such as $3x - 5$ and $3(x + h) - 5$ are considered algebraic expressions. They are used to describe and manipulate values that depend on one or more variables.

**Variables:** Symbols like $x$ and $h$ represent numbers that can vary.
**Constants:** Numbers like 3 and -5 that do not change within the expression.
**Operators:** Symbols such as +, -, and * that dictate the operation to be performed.The expression $3x - 5$ indicates a process: multiplying $x$ by 3 and then subtracting 5. It's important to understand and identify these components as they play a crucial role in the manipulation and simplification of expressions in algebra.

Function Evaluation

Function evaluation involves determining the output of a function for particular input values. In other words, it's about finding what the function results in when you replace the variable with a specific value.In this exercise, we start with the function $P(x) = 3x - 5$. The task requires evaluating $P(x+h)$. This means:

Substitute $x + h$ in place of $x$ in the function.
Perform the arithmetic operations as indicated.

When we substitute $x = x + h$ into $P(x)$, the expression becomes $3(x + h) - 5$. Evaluating this gives $3x + 3h - 5$. This process demonstrates evaluating a polynomial function at a shifted variable. Understanding evaluation is essential for manipulating functions and working through algebraic problems efficiently.

Substitution Method

The substitution method is a technique used to replace a variable in an expression or an equation with another expression. This strategy allows for the simplification or solving of equations more easily.Here's how it played out in the exercise:

We substituted $x + h$ wherever $x$ appeared in $P(x) = 3x - 5$.
In the expression $3(x + h) - 5$, replace each $x$ with $x + h$, leading to $3x + 3h - 5$.

Then, bringing back the original $P(x)$ or $(3x - 5)$, and adding it to the new expression demonstrates using substitution in tandem with function evaluation.Using substitution, you see how different values affect the outcome of expressions, and learn to manipulate these expressions correctly. This is vital in handling complex algebra problems where multiple variables and substitutions come into play.

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91影视

Let \(P(x)=3 x-5 .\) Find each value. $$P(x+h)+P(x)$$

Short Answer

Step by step solution

Substitute \(x+h\) into \(P(x)\)

Determine \(P(x)\)

Add \(P(x + h)\) and \(P(x)\)

Key Concepts

Algebraic Expressions

Function Evaluation

Substitution Method

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