Problem 37 - Algebra Form and Function

Question

Write the polynomials in exercises in standard form $$a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ What are the values of the coefficients \(a_{0}, a_{1}, \ldots, a_{n} ?\) Give the degree of the polynomial. $$ (x+1)^{3} $$

Step-by-Step Solution

Answer

Answer: The coefficients of the simplified polynomial are \(a_0=1\), \(a_1=3\), \(a_2=3\), and \(a_3=1\). The degree of the polynomial is \(3\).

Step 1: Expanding the given polynomial expression

To expand the expression \((x+1)^3\), apply the binomial theorem. Using the binomial theorem, we can rewrite it as: $$(x+1)^3=\binom{3}{0}x^3\cdot1^0+\binom{3}{1}x^2\cdot1^1+\binom{3}{2}x^1\cdot1^2+\binom{3}{3}x^0\cdot1^3$$ where \(\binom{n}{k}=\frac{n!}{k!(n-k)!}\) is the binomial coefficient.

Step 2: Simplifying the expression

Now let's simplify the polynomial by calculating the binomial coefficients and performing the multiplication with the powers of \(x\): $$ \begin{aligned} (x+1)^3 &= \frac{3!}{0!(3-0)!}x^3\cdot1^0+\frac{3!}{1!(3-1)!}x^2\cdot1^1+\frac{3!}{2!(3-2)!}x^1\cdot1^2+\frac{3!}{3!(3-3)!}x^0\cdot1^3 \\ &= 1\cdot x^3 + 3x^2 + 3x + 1 \end{aligned} $$

Step 3: Identifying the coefficients

Looking at the simplified polynomial expression, we can identify the coefficients \(a_0\), \(a_1\), \(a_2\), and \(a_3\) from the standard form \(a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}\): $$1\cdot x^3+3\cdot x^2+3\cdot x+1$$ In this case, \(a_0=1\), \(a_1=3\), \(a_2=3\), and \(a_3=1\).

Step 4: Determining the degree of the polynomial

The degree of a polynomial is the highest power of \(x\) present in it. The expanded polynomial is: $$1\cdot x^3+3\cdot x^2+3\cdot x+1$$ Here, the highest power of \(x\) is \(3\). Thus, the degree of the polynomial is \(3\).

Key Concepts

Binomial Theorem

The binomial theorem is a powerful tool in algebra. It provides a formula for expanding binomials raised to any positive integer power. A binomial is an expression with two terms, like \((a + b)^n\). The theorem states that:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Here, \(\binom{n}{k}\) is the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\). This coefficient tells us how many ways we can choose \(k\) elements from a total of \(n\).
In our exercise, we expanded \((x+1)^3\) using this theorem. We re-wrote it as:\(\binom{3}{0}x^3\cdot1^0+\binom{3}{1}x^2\cdot1^1+\binom{3}{2}x^1\cdot1^2+\binom{3}{3}x^0\cdot1^3\),resulting in the final polynomial \(x^3 + 3x^2 + 3x + 1\). Using the binomial theorem makes expanding a straightforward process, even if the power \(n\) is large.
The binomial theorem is commonly used in combinatorics, calculus, and even probability theory. Understanding it helps simplify complex algebraic tasks into more manageable parts.

Degree of a Polynomial

The degree of a polynomial is simply the highest power of the variable \(x\) in the expression. Each term in a polynomial has a degree, and the degree of the polynomial is determined by the term with the largest degree. This concept is crucial because it gives us insights into the polynomial's behavior, especially in graphing.In the expanded form of \((x+1)^3\): \(x^3 + 3x^2 + 3x + 1\),The term with the highest power of \(x\) is \(x^3\). Thus, the degree of the polynomial is 3. This tells us that the polynomial is of third degree, or cubic.
Understanding the degree helps in:

• Determining the polynomial's end behavior in graphical representations.
• Characterizing the number of roots, typically equal to its degree.
• Indicating possible turning points in graph plots since a third-degree polynomial can have up to 2 turning points.
  

Recognizing the degree of a polynomial allows us to predict its properties and solutions more efficiently.

Standard Form of a Polynomial

A polynomial is in its standard form when its terms are written in descending order of the power of \(x\). Each term consists of a coefficient and a variable raised to a non-negative integer exponent.For example, consider the polynomial derived from \((x+1)^3\),\(1 \cdot x^3 + 3 \cdot x^2 + 3 \cdot x + 1\).Here, the terms are arranged from \(x^3\) to the constant term \(x^0\). This is the standard form of the polynomial:

• It makes it easy to identify the leading term, which influences the end behavior of the function.
• The coefficients \(a_n\) down to \(a_0\) become clear, enabling simpler calculations for evaluations and operations.
  

Understanding a polynomial's standard form helps in analyzing its graph and solving problems related to its function.The standard form assists in computational ease and clarity, regularizing how we approach polynomial equations.