Problem 10 - Algebra Form and Function

Question

Give the constant term, \(a_{0}\). $$ t(t-1)(t-2) $$

Step-by-Step Solution

Answer

Answer: The constant term \(a_0\) of the given polynomial is 0.

Step 1: Expand the product - First two factors

First, expand the product of the first two factors \((t)(t - 1)\): $$ (t)(t - 1) = t^2 - t $$ Now, we have the product \((t^2 - t)(t - 2)\).

Step 2: Expand the product - Second and third factors

We now expand the product of the second and third factors \((t^2 - t)(t - 2)\): $$ (t^2 - t)(t - 2) = t^3 - 2t^2 - t^2 + 2t = t^3 - 3t^2 + 2t $$

Step 3: Identify the constant term

After expanding the product, we have the polynomial \(t^3 - 3t^2 + 2t\). Here, the term without \(t\) is the constant term. Therefore, the constant term, \(a_0\), is 0, as there is no constant term in this polynomial. So, the constant term \(a_0\) of the given polynomial is: $$ a_0 = 0 $$

Key Concepts

Constant Term

In any polynomial expression, the constant term is the term that has no variable attached to it. This means it does not change as the variable changes, making it 'constant' regardless of the variable's value.
For example, in the simple expression \(x^2 + 3x + 5\), the constant term is 5.
To identify the constant term in a given polynomial, look at the terms and find the one without any variable part.When expanding polynomials through multiplication, sometimes no constant term appears. Like in the initial polynomial expression \(t^3 - 3t^2 + 2t\), there is no term left without \(t\), indicating the constant term, \(a_0\), is zero.
Remember, a zero constant means there's nothing left if all the variable terms were set to zero. However, this doesn't change the polynomial's overall structure.

Polynomial Expression

A polynomial expression is a mathematical expression composed of variables, coefficients, and non-negative integer powers (exponents) of variables. Polynomials are expressed in the form, such as \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where:

• Each term involves a coefficient (\(a_i\)) and a power of the variable (\(x^i\)).
• The powers of the variables indicate the order of the polynomial.
• The coefficients are numbers that can be positive, negative, or zero.

The given expression, \(t(t-1)(t-2)\), must be expanded to identify the polynomial's structure.
This form shows how different factors combine into a single polynomial form with expanded terms.
Overall, understanding polynomials involves recognizing how they're built from simpler pieces, each adding complexity depending on their order and coefficients.

Multiplication of Binomials

Multiplying binomials is a key technique in expanding polynomial expressions. A binomial is simply a polynomial with two terms, like \((x + y)\).
The product of two binomials can be found using the distributive property, commonly remembered through the acronym FOIL (First, Outer, Inner, Last).

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outermost terms in the expression.
• Inner: Multiply the innermost terms in the expression.
• Last: Multiply the last terms in each binomial.

To expand \((t^2 - t)(t-2)\), apply the distributive method:\[(t^2)\cdot(t) + (t^2)\cdot(-2) + (-t)\cdot(t) + (-t)\cdot(-2)\]This becomes \(t^3 - 2t^2 - t^2 + 2t\).
These terms can be simplified further into a complete polynomial \(t^3 - 3t^2 + 2t\), showcasing how binomials multiply into higher-degree polynomials.