Problem 74 - Algebra Form and Function

Question

Find the leading term and constant term of (a) \((x-1)^{2}\) (b) \((x-1)^{3}\) (c) \((x-1)^{4}\)

Step-by-Step Solution

Answer

Question: Find the leading term and constant term of the following expressions: (a) (x-1)^2 (b) (x-1)^3 (c) (x-1)^4 Answer: (a) Leading term: x^2, Constant term: 1 (b) Leading term: x^3, Constant term: -1 (c) Leading term: x^4, Constant term: 1

Step 1: (a) Finding the leading term and constant term of \((x-1)^{2}\)

First, we need to expand \((x-1)^{2}\). We can use the binomial theorem, which states that for any natural number n, \(\sum_{k=0}^{n}\binom{n}{k} x^{n-k} (y)^{k} = (x+y)^n\). In our case, x = x, y = -1, and n = 2. \(\binom{2}{0}x^2(-1)^0 + \binom{2}{1}x^1(-1)^1 + \binom{2}{2}x^0(-1)^2 = x^2-2x+1\). The leading term is \(x^2\) and the constant term is 1.

Step 2: (b) Finding the leading term and constant term of \((x-1)^{3}\)

Similar to (a), we will now use the binomial theorem to expand \((x-1)^{3}\), using x = x, y = -1, and n = 3. \(\binom{3}{0}x^3(-1)^0 + \binom{3}{1}x^2(-1)^1 + \binom{3}{2}x^1(-1)^2 + \binom{3}{3}x^0(-1)^3= x^3-3x^2+3x-1\) The leading term is \(x^3\) and the constant term is -1.

Step 3: (c) Finding the leading term and constant term of \((x-1)^{4}\)

Finally, we use the binomial theorem one more time to expand \((x-1)^{4}\), using x = x, y = -1, and n = 4. \(\binom{4}{0}x^4(-1)^0 + \binom{4}{1}x^3(-1)^1 + \binom{4}{2}x^2(-1)^2 + \binom{4}{3}x^1(-1)^3 + \binom{4}{4}x^0(-1)^4= x^4-4x^3+6x^2-4x+1\) The leading term is \(x^4\) and the constant term is 1.

Key Concepts

Leading Term

When exploring polynomial expressions, the leading term is a crucial concept. It refers to the term in the polynomial with the highest degree of the variable, which often dictates the polynomial's overall behavior as the variable grows larger.
For example, in the polynomial expansion of \(x-1\)^2 = x^2 - 2x + 1, the leading term is \(x^2\).
It's the term that contains the highest power of \(x\), specifically \(x^2\) in this scenario, which informs us of the polynomial's degree.

• The leading term helps to predict the end behavior of the polynomial graph.
• When comparing polynomials, their leading terms are often used to determine which grows faster as \(x\) increases.
• In our example, \(x^3\) serves as the leading term for both \(x-1\)^3 and \(x^4\) for \(x-1\)^4 indicating the degrees of these polynomials.

By identifying leading terms, you gain insight into the polynomial's fundamental properties and behavior in analytical and graph form.

Constant Term

The constant term is the part of the polynomial that does not have any variable attached to it. In our polynomial expansions, it remains unaltered by changes in the variable.
For instance, in the expansion of \(x-1\)^2 = x^2 - 2x + 1, the constant term is \(1\).
It stands alone without an \(x\) factor.

• It provides the y-intercept of the polynomial graph, where the function intersects the y-axis.
• In mathematical expressions, it can affect the balance and solution of equations involving polynomials.
• In the examples provided, the constant terms are 1 for both \(x-1\)^2 and \(x-1\)^4, while \(x-1\)^3 has a constant term of -1.

Understanding the constant term is pivotal when calculating roots or intercepts and interpreting static values in polynomial functions.

Polynomial Expansion

Polynomial expansion is the process of expressing a polynomial in its standard form, usually using the binomial theorem for complex expressions. This theorem helps expand powers of binomial expressions such as \(x-1\)^n.
The binomial theorem states: \[ (x+y)^n = \sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^k \].

• This setup uses binomial coefficients \(\binom{n}{k}\) to distribute terms expansively.
• As observed in the exercise, expanding \(x-1\)^2 using the theorem, we arrive at \(x^2 - 2x + 1\).
• Each term emerges from factoring with corresponding coefficients \(\binom{n}{k}\), which count possible ways of selecting items without regard to order.

The binomial theorem not only aids in polynomial expansion but is integral in probability, statistics, and algebra to simplify expressions and solve intricate problems.