Problem 12 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Let \(P(x)=2 x^{3}-7 x^{2}+5\) and \(Q(x)=x^{2}-x\). Use algebra to compute \(Q(P(x))\). You may conclude (correctly) from this exercise that the composition of two polynomials is always a polynomial.

Step-by-Step Solution

Answer

The composition \(Q(P(x)) = 4x^6 - 28x^5 + 49x^4 + 18x^3 - 63x^2 + 20\) is a polynomial.

Step 1: Understand the Expression

To find \(Q(P(x))\), recognize that it is the composition of the polynomial \(Q(x) = x^2 - x\) with \(P(x) = 2x^3 - 7x^2 + 5\). This means substituting \(P(x)\) into \(Q(x)\).

Step 2: Substitute P(x) into Q(x)

Substitute \(P(x) = 2x^3 - 7x^2 + 5\) for \(x\) in \(Q(x)\). This gives us \(Q(P(x)) = (2x^3 - 7x^2 + 5)^2 - (2x^3 - 7x^2 + 5)\).

Step 3: Expand the Squared Term

Expand \((2x^3 - 7x^2 + 5)^2\). Use the formula \((a-b+c)^2 = a^2 - 2ab + 2ac + b^2 - 2bc + c^2\) where \(a = 2x^3\), \(b = 7x^2\), and \(c = 5\).

Step 4: Calculate Each Part of the Expansion

Calculate:1. \((2x^3)^2 = 4x^6\)2. \((-7x^2)^2 = 49x^4\)3. \(5^2 = 25\)4. \(-2 \times 2x^3 \times 7x^2 = -28x^5\)5. \(2 \times 2x^3 \times 5 = 20x^3\)6. \(-2 \times 7x^2 \times 5 = -70x^2\)

Step 5: Combine the Expansion Terms

After obtaining the individual terms from the expansion in Step 4, combine them to get: \[4x^6 - 28x^5 + 49x^4 + 20x^3 - 70x^2 + 25\].

Step 6: Subtract P(x) from the Expansion

Subtract \(P(x) = 2x^3 - 7x^2 + 5\) from \(4x^6 - 28x^5 + 49x^4 + 20x^3 - 70x^2 + 25\). Distribute the negative sign and combine like terms:\[-2x^3 + 7x^2 - 5\].

Step 7: Final Computation

Combine like terms from the result of Step 6:\[4x^6 - 28x^5 + 49x^4 + 18x^3 - 63x^2 + 20\].

Step 8: Conclusion on Polynomials Composition

The resulting expression \(4x^6 - 28x^5 + 49x^4 + 18x^3 - 63x^2 + 20\) is a polynomial, thereby showing that the composition of two polynomials is indeed a polynomial.

Key Concepts

Algebraic Expansion

Algebraic expansion is an essential concept in algebra that involves the distribution of terms within an expression. When expanding expressions, especially polynomial compositions, the goal is to rewrite the expression in a more "expanded" form. This helps in identifying like terms and simplifies further operations. Consider the case of

• expanding \( (2x^3 - 7x^2 + 5)^2 \) as seen in the exercise.
• It involves applying the binomial theorem or distribution methods to multiply the terms completely.

For example, using \(a^2 - 2ab + b^2\) formula, where you identify parts of the expression: \(a = 2x^3\), \(b = 7x^2\), and \(c = 5\). This requires multiplying each possible combination of terms:

• \( (2x^3)^2 = 4x^6\)
• \( (-7x^2)^2 = 49x^4 \)
• \((-2 \times 2x^3 \times 7x^2 = -28x^5)\)

The algebraic expansion reveals the true structure of a complex polynomial, breaking it down into all its summarized parts.

Polynomial Functions

Polynomial functions are fundamental in algebra, typically expressed in the standard form of \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\). These functions are characterized by

• their degree, which is dictated by the highest power of \(x\),
• and the coefficients, which are the constants \(a_n\).

In our exercise, we dealt with two polynomial functions:

• \(P(x) = 2x^3 - 7x^2 + 5\)
• \(Q(x) = x^2 - x\)

Understanding these individual functions is crucial when performing operations like composition, as it allows us to comprehend the impact of combining them through substitution. The nature of polynomial functions ensures that their degree and behavior are retained, even when manipulated into more complex expressions.

Mathematical Substitution

Mathematical substitution is an operation of replacing expressions with their equivalent values or functions. In polynomial compositions like \(Q(P(x))\), substitution plays a vital role in transforming one expression into another. Here’s the idea:

• We take the entire expression for \(P(x)\), which is \(2x^3 - 7x^2 + 5\),
• and substitute it directly into \(Q(x)\).

This changes the single variable \(x\) in \(Q(x)\) with the complex polynomial from \(P(x)\), resulting in the new expression \(Q(P(x)) = (2x^3 - 7x^2 + 5)^2 - (2x^3 - 7x^2 + 5)\).
Substitution helps establish the relationship between two functions and allows for their combination into a single polynomial function. Through substitution, new and more elaborate functions are formed, extending the possibilities of algebraic exploration and problem-solving.