Problem 51 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The expression evaluates to \(4x^4 - 9x^3 + 28x^2 - 33x + 20\).

Step 1: Substitute into p(x)

First, substitute \(x^2 + 1\) into the function \(p(x) = 2x^2 - 5x + 4\). This means replacing every \(x\) in \(p(x)\) with \(x^2 + 1\). So we calculate: \(p(x^2 + 1) = 2(x^2 + 1)^2 - 5(x^2 + 1) + 4\).

Step 2: Simplify p(x^2 + 1)

Expand \((x^2 + 1)^2\) and simplify:1. \((x^2 + 1)^2 = x^4 + 2x^2 + 1\).2. Substitute into \(p(x^2 + 1)\):\[p(x^2 + 1) = 2(x^4 + 2x^2 + 1) - 5(x^2 + 1) + 4\]Expand: \[= 2x^4 + 4x^2 + 2 - 5x^2 - 5 + 4\]Combine like terms: \[= 2x^4 - x^2 + 1\].

Step 3: Substitute into r(x)

Substitute \(x - 1\) into the function \(r(x) = 3x^3 - x^2 - 2\). Replace \(x\) with \(x-1\): \(r(x-1) = 3(x-1)^3 - (x-1)^2 - 2\).

Step 4: Simplify r(x-1)

Expand \((x-1)^3\) and \((x-1)^2\):1. \((x-1)^3 = x^3 - 3x^2 + 3x - 1\).2. \((x-1)^2 = x^2 - 2x + 1\).Substitute these back:\[r(x-1) = 3(x^3 - 3x^2 + 3x - 1) - (x^2 - 2x + 1) - 2\]Expand: \[= 3x^3 - 9x^2 + 9x - 3 - x^2 + 2x - 1 - 2\]Combine like terms: \[= 3x^3 - 10x^2 + 11x - 6\].

Step 5: Combine Results

Now we plug back into the expression \(2[p(x^2 + 1)] - 3r(x-1)\):1. Multiply \(p(x^2 + 1)\) by 2:\[2(2x^4 - x^2 + 1) = 4x^4 - 2x^2 + 2\]2. Multiply \(r(x-1)\) by 3:\[3(3x^3 - 10x^2 + 11x - 6) = 9x^3 - 30x^2 + 33x - 18\]Finally:\[2(p(x^2 + 1)) - 3r(x-1) = (4x^4 - 2x^2 + 2) - (9x^3 - 30x^2 + 33x - 18)\]Combine like terms into:\[= 4x^4 - 9x^3 + 28x^2 - 33x + 20\]

Step 6: Verify and Finalize

Double-check each calculation to ensure accuracy. Verify through careful expansion and simplification. The final expression is:\[= 4x^4 - 9x^3 + 28x^2 - 33x + 20\]

Key Concepts

Polynomial Evaluation

Polynomial evaluation involves calculating the value of a polynomial function for a given input. This means plugging in specific numbers or expressions for the variable and simplifying to find the resulting value.
For instance, in the problem, we have the polynomial function \(p(x) = 2x^2 - 5x + 4\). To evaluate the polynomial when \(x = x^2 + 1\), you substitute \(x^2 + 1\) into every place you see \(x\) in \(p(x)\).
This results in the expression \(p(x^2 + 1) = 2(x^2 + 1)^2 - 5(x^2 + 1) + 4\). The next step will be expanding and simplifying this expression to find its simplest form.

Function Composition

Function composition is the process of replacing the input of one function with another function. This can be thought of as a function within a function.
In the exercise, we see this when \(x^2 + 1\) replaces \(x\) in \(p(x)\), meaning we're evaluating \(p\) at the output of another function, \(x^2 + 1\).
Similarly, we have \(r(x) = 3x^3 - x^2 - 2\), which is composed with \(x - 1\). This requires substituting \(x - 1\) where \(x\) appears in \(r(x)\), resulting in \(r(x-1) = 3(x-1)^3 - (x-1)^2 - 2\).
Function composition is key in forming new functions from existing ones by combining their operations.

Expression Simplification

Expression simplification is an essential step in working with polynomials, where the goal is to simplify an expression to its simplest form. It involves expanding products, combining like terms, and performing basic arithmetic operations.
In our example, after substituting \(x^2 + 1\) into \(p(x)\), we ended up with a longer expression \(2(x^4 + 2x^2 + 1) - 5(x^2 + 1) + 4\). To simplify, we follow these steps:

• Expand any squares, such as \((x^2 + 1)^2 = x^4 + 2x^2 + 1\).
• Combine like terms for a simplified polynomial, resulting in \(2x^4 - x^2 + 1\).

Similarly, for \(r(x-1)\), we took a complex expression and simplified it to \(3x^3 - 10x^2 + 11x - 6\), highlighting the importance of careful operations to isolate core polynomial terms.

Polynomial Expansion

Polynomial expansion involves multiplying and simplifying variables and constants to open up expressions, particularly useful when dealing with powers or products.
For example, expanding \((x-1)^3\) in \(r(x-1)\) involves using the distributive property to achieve \(x^3 - 3x^2 + 3x - 1\).
Once expanded, terms can be combined through simplification. Another example is expanding \((x^2 + 1)^2\) to determine its full expression, \(x^4 + 2x^2 + 1\).
Expansion helps in making each term explicit, allowing for effective combination and simplification, essential to solving polynomial expressions efficiently.