Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ \left(x^{2}+1\right)\left(x^{2}-16\right) $$

Step-by-Step Solution

Answer

The polynomial is \(x^4 - 15x^2 - 16\).

Step 1: Distribute the First Term

Multiply the first term of the first binomial, \(x^2\), with both terms of the second binomial, \((x^2 - 16)\). This step gives us two terms: \(x^2 \times x^2 = x^4\) and \(x^2 \times (-16) = -16x^2\).

Step 2: Distribute the Second Term

Multiply the second term of the first binomial, \(+1\), with both terms of the second binomial, \((x^2 - 16)\). This yields two more terms: \(1 \times x^2 = x^2\) and \(1 \times (-16) = -16\).

Step 3: Combine the Terms

Combine all results from the previous steps: \(x^4 + (-16x^2) + x^2 - 16\).

Step 4: Simplify the Polynomial

Combine like terms. In this expression, the like terms are the \(x^2\) terms: \(-16x^2 + x^2 = -15x^2\). Hence, the polynomial becomes: \(x^4 - 15x^2 - 16\).

Key Concepts

Binomial Expressions

A binomial expression is a polynomial with exactly two terms. Each term in a binomial can be a number, a variable, or a product of numbers and variables. In the exercise, we dealt with two binomial expressions: \(x^2 + 1\) and \(x^2 - 16\). Understanding the structure of a binomial is crucial as it helps in identifying and applying the correct operations.

Some examples of binomial expressions include:

• \(a + b\)
• \(3x - 4y\)
• \(x^3 + 7\)

These expressions are foundational as they often appear in polynomial multiplication tasks. Recognizing and understanding binomials is the first step in simplifying or multiplying them.

Distributive Property

The distributive property is a fundamental rule in algebra. It allows you to multiply a sum by multiplying each addend separately and then add the results. In our exercise, the distributive property was used to multiply each term in the first binomial \((x^2 + 1)\) by each term in the second binomial \((x^2 - 16)\).

Here's how it works:

• Multiply the first term \(x^2\) by every term in the second binomial \((x^2 \) and \(-16)\) to get \(x^4\) and \(-16x^2\).
• Next, multiply the second term \(+1\) by each term in the second binomial to get \(x^2\) and \(-16\).

These steps show how each component of one binomial interacts with each component of the other, breaking the multiplication task into easier, more manageable pieces. This property is especially helpful when dealing with complex expressions.

Combining Like Terms

Combining like terms is a method used to simplify polynomial expressions. Like terms are terms in a polynomial that have the same variable raised to the same power.

For example, in the polynomial resulting from our exercise \(x^4 + (-16x^2) + x^2 - 16\), the like terms are \(-16x^2\) and \(+x^2\). Combining these terms involves summing their coefficients:

• \(-16x^2 + x^2 = -15x^2\)

This step helps in reducing the polynomial to its simplest form by consolidating terms, reducing complexity, and paving the way for an easy-to-understand final expression. Knowing how to identify and combine like terms is essential when working with any polynomial.

Simplifying Polynomials

Simplifying polynomials involves performing operations to rewrite the expression in its most reduced form, without changing its value.

This is the final step in our exercise, which took the expanded polynomial \(x^4 - 15x^2 - 16\), and confirmed that there were no more like terms to combine or reduce further.

• Simplification makes it easier to understand and work with a polynomial.
• It helps in solving equations, graphing functions, and performing further calculations.

When simplifying, make sure all like terms are combined and the polynomial is free from unnecessary parentheses or terms. This process not only applies to multiplication tasks like the one we're discussing but also to polynomial addition, subtraction, and division.