Problem 26 - Algebra 2

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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 \(r(x)=x^3 + x + 1\).

Step 3: 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 \((x+1)(x+1)(x+1)\).

Step 4: Simplify Expression

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

Step 5: Combine Like Terms

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

Key Concepts

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.