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Chapter 6: Problem 26

If \(p(x)=3 x^{2}-2 x+5\) and \(r(x)=x^{3}+x+1,\) find each value. \(r(x+1)\)

Short Answer

Expert verified
\(r(x+1) = x^3 + 3x^2 + 4x + 2\)

Step by step solution

01

Understand the Problem

We need to evaluate the polynomial function \(r(x)\) at \(x+1\). The function is given by \(r(x)=x^{3}+x+1\). We will replace \(x\) with \(x+1\) in the function.
02

Substitute \(x+1\) into \(r(x)\)

Write the expression \(r(x+1) = (x+1)^3 + (x+1) + 1\). Substitute \(x+1\) for \(x\) in each occurrence in the polynomial \(r(x)=x^3 + x + 1\).
03

Expand \((x+1)^3\)

The term \((x+1)^3\) expands to \(x^3 + 3x^2 + 3x + 1\) using the binomial theorem or by manually expanding \((x+1)(x+1)(x+1)\).
04

Simplify Expression

Combine all terms in the expression: \(x^3 + 3x^2 + 3x + 1 + x + 1\).
05

Combine Like Terms

Add the like terms together: \(x^3 + 3x^2 + 4x + 2\). This is the simplified form of \(r(x+1)\).

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Key Concepts

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

Binomial Theorem
The Binomial Theorem is a fundamental concept in algebra used to expand expressions that are raised to a power, particularly binomials. A binomial is an expression containing two terms, such as \( (a + b) \). When raised to a power, say \( n \), we are interested in expanding it to a sum of terms formed by the coefficients and powers of \( a \) and \( b \).

To expand \( (x + 1)^3 \), as our original problem requires, the Binomial Theorem tells us:
  • Each term in the expansion includes a combination of powers of the two terms \( x \) and \( 1 \).
  • The powers of \( x \) decrease from 3 to 0, while the powers of 1 increase from 0 to 3.
  • The coefficients of each term are determined by the binomial coefficients, which follow patterns in Pascal's Triangle or can be calculated using combinations \( \binom{n}{k} \).
For our specific expansion:\((x + 1)^3 = x^3 + 3x^2 + 3x + 1\).

Understanding this theorem helps make expanses less daunting and ensures no mistakes in the distribution process.
Polynomial Expansion
Polynomial Expansion is the method of expanding expressions from their factored form to a fully distributed format. This concept is essential when managing polynomial expressions to see each term distinctly.

In the given exercise, the polynomial \( r(x) = (x+1)^3 + (x+1) + 1 \) needs to be expanded. Firstly, we focus on expanding \((x+1)^3\), as shown using the Binomial Theorem. Then, add the remaining individual linear polynomial terms:
  • The term \((x+1)\) presents itself directly as \( x + 1 \).
  • The constant 1 remains unchanged.
Combining these with the expansion results in:\[r(x+1) = (x^3 + 3x^2 + 3x + 1) + x + 1\].

This expansion shows its components clearly, making it much easier to analyze and simplify further.
Combining Like Terms
Combining Like Terms is an important step in simplifying a polynomial to its simplest form. Like Terms refer to terms that share the same variable raised to the same power, allowing them to be easily combined by addition or subtraction.

After expanding \( r(x+1) = x^3 + 3x^2 + 3x + 1 + x + 1 \), we identify the process of combining like terms to simplify:
  • \(x^3\) stands alone as there are no other such cubed terms.
  • \(3x^2\) similarly remains unchanged, being the sole squared term.
  • Linear terms are combined: \(3x + x = 4x\).
  • Constant terms simply add up: \(1 + 1 = 2\).
These steps lead to the simplified polynomial:\[x^3 + 3x^2 + 4x + 2\].

This operation streamlines polynomial expressions and is a key tool in algebra that prepares the polynomial for further operations or evaluation.

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Most popular questions from this chapter

PERSONAL FINANCE For Exercises \(38-41,\) use the following information. Zach has purchased some home theater equipment for \(\$ 2000,\) which he is financing through the store. He plans to pay \(\$ 340\) per month and wants to have the balance paid off after six months. The formula \(B(x)=2000 x^{6}-\) 340\(\left(x^{5}+x^{4}+x^{3}+x^{2}+x+1\right)\) represents his balance after six months if \(x\) represents 1 plus the monthly interest rate (expressed as a decimal). How would the formula change if Zach wanted to pay the balance in five months?

Use synthetic substitution to find \(g(3)\) and \(g(-4)\) for each function. $$ g(x)=2 x^{3}-8 x^{2}-2 x+5 $$

REVIEW. The total area of a rectangle is \(25 a^{4}-16 b^{2} .\) Which factors could represent the length times width? \(\mathbf{F}\left(5 a^{2}+4 b\right)\left(5 a^{2}+4 b\right)\) \(\mathbf{G}\left(5 a^{2}+4 b\right)\left(5 a^{2}-4 b\right)\) \(\mathbf{H}(5 a-4 b)(5 a-4 b)\) \(\mathbf{J}(5 a+4 b)(5 a-4 b)\)

Use synthetic substitution to find \(g(3)\) and \(g(-4)\) for each function. $$ g(x)=x^{4}-6 x-8 $$

ENGINEERING. For Exercises 32 and \(33,\) use the following information. When a certain type of plastic is cut into sections, the length of each section determines its strength. The function \(f(x)=x^{4}-14 x^{3}+69 x^{2}-140 x+100\) can describe the relative strength of a section of length \(x\) feet. Sections of plastic \(x\) feet long, where \(f(x)=0,\) are extremely weak. After testing the plastic, engineers discovered that sections 5 feet long were extremely weak. Show that \(x-5\) is a factor of the polynomial function.
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