Problem 40 - Algebra 2

Question

If \(p(x)=3 x^{2}-2 x+5\) and \(r(x)=x^{3}+x+1,\) find each value. 2\([p(x+4)]\)

Step-by-Step Solution

Answer

The result is \(6x^2 + 44x + 90\).

Step 1: Calculate p(x+4)

First, we need to find the expression for \( p(x+4) \). We start by substituting \( (x+4) \) into the original polynomial \( p(x) = 3x^2 - 2x + 5 \). So, \( p(x+4) = 3(x+4)^2 - 2(x+4) + 5 \).

Step 2: Expand (x+4)^2

Now we need to expand \( (x+4)^2 \). \((x+4)^2 = x^2 + 8x + 16\).

Step 3: Substitute and simplify

Substitute \( x^2 + 8x + 16 \) into the expression for \( p(x+4) \): \[ p(x+4) = 3(x^2 + 8x + 16) - 2x - 8 + 5 \]. Simplify to obtain \( p(x+4) = 3x^2 + 24x + 48 - 2x + 5 - 8 \).

Step 4: Combine like terms

Now combine the like terms in the equation: \( p(x+4) = 3x^2 + 22x + 45 \).

Step 5: Double p(x+4)

Finally, calculate \( 2[p(x+4)] \), which gives us \( 2(3x^2 + 22x + 45) = 6x^2 + 44x + 90 \).

Key Concepts

Polynomial Expressions

A polynomial expression is a combination of variables and coefficients, involving operations such as addition, subtraction, and multiplication. In our exercise, we are given two polynomial expressions: \( p(x) = 3x^2 - 2x + 5 \) and \( r(x) = x^3 + x + 1 \). Polynomials are classified based on their degree, which is determined by the highest power of the variable present in the expression.

• A linear polynomial has a degree of 1.
• A quadratic polynomial, like \( p(x) \), has a degree of 2.
• A cubic polynomial, like \( r(x) \), has a degree of 3.

Understanding polynomial expressions is essential for evaluating and manipulating them effectively in mathematical calculations.

Substitution in Polynomials

Substitution in polynomials involves replacing the variable \( x \) with a different expression or value. This is essential when calculating specific values of the polynomial. In the case of our exercise, we were tasked to substitute \( x \) with \( x+4 \) in the polynomial \( p(x) \). By substitution, the polynomial transforms, allowing for further computation and manipulation.
To substitute, you replace every occurrence of the variable \( x \) with \( x+4 \), resulting in a new polynomial expression \( p(x+4) = 3(x+4)^2 - 2(x+4) + 5 \). Substitution is a fundamental technique used in algebra to explore the behavior of polynomials under changing conditions.

Expanding Binomials

Expanding binomials, such as \( (x+4)^2 \), means to distribute the expression to eliminate parentheses. This process involves using the distributive property and finding the product of each term. For \( (x+4)^2 \), you calculate it as:\((x+4)(x+4)\), which leads to:
\[ (x+4)(x+4) = x^2 + 4x + 4x + 16 \].
Combining like terms, it simplifies to \( x^2 + 8x + 16 \).
Expanding is critical to perform further algebraic operations and accurately arrange polynomial terms for simplification.

Simplifying Expressions

Simplifying expressions involves combining like terms to reduce a polynomial to its simplest form. Like terms have the same power of \( x \) and can be added or subtracted. In our exercise, once all the variable substitutions and expansions were completed, a new expression \( 3(x^2 + 8x + 16) - 2x + 5 - 8 \) was simplified.
The steps involved:

• Multiply each term by its coefficient.
• \(3(x^2 + 8x + 16)\) becomes \(3x^2 + 24x + 48\).
• Combine with the other terms to get \(3x^2 + 22x + 45\).

Simplification helps in reducing the complexity of calculations, making it easier to perform further operations like finding values for different substitutions.