Problem 48 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ \left(1+x+x^{2}\right)\left(1-x+x^{2}\right) $$

Step-by-Step Solution

Answer

The simplified result is \(1 + x^4\).

Step 1: Distribute the First Term

Take the first term from the first polynomial, which is 1, and distribute it across the second polynomial \((1-x+x^2)\). This results in \(1 imes 1 - 1 imes x + 1 imes x^2 = 1 - x + x^2\).

Step 2: Distribute the Second Term

Take the second term from the first polynomial, which is \(x\), and distribute it across the second polynomial \((1-x+x^2)\). This results in \(x imes 1 - x imes x + x imes x^2 = x - x^2 + x^3\).

Step 3: Distribute the Third Term

Take the third term from the first polynomial, which is \(x^2\), and distribute it across the second polynomial \((1-x+x^2)\). This results in \(x^2 imes 1 - x^2 imes x + x^2 imes x^2 = x^2 - x^3 + x^4\).

Step 4: Combine Like Terms

Now combine all the results from steps 1, 2, and 3: \((1 - x + x^2) + (x - x^2 + x^3) + (x^2 - x^3 + x^4)\). Simplify by combining like terms: \(1 + 0x + 0x^2 + 0x^3 + x^4\). Only the constant term (1) and the term with \(x^4\) remain, as others cancel each other.

Key Concepts

Distributive Property

In polynomial multiplication, the distributive property is a key step that allows us to tackle each part of the multiplication process individually. The distributive property states that for any numbers or variables, the expression \(a(b+c)\) is equal to \(ab + ac\). It helps in breaking down the multiplication of polynomials into simpler, manageable parts.

When you multiply \((1+x+x^2)(1-x+x^2)\), you distribute each term of the first polynomial over every term in the second polynomial. This means:

• First, you multiply the constant 1 from the first polynomial across each term in the second polynomial: \((1)(1) - (1)(x) + (1)(x^2)\).
• Then, you do the same with the term \(x\): \((x)(1) - (x)(x) + (x)(x^2)\).
• Finally, you distribute the \(x^2\) term: \((x^2)(1) - (x^2)(x) + (x^2)(x^2)\).

By distributing each term, you ensure no part of the multiplication is overlooked, making it easier to later combine like terms.

Combining Like Terms

After distributing the terms, the next step is to collect or combine like terms. Like terms are terms that have the same variable raised to the same power. In contrast to terms that aren't like, these can be added or subtracted.

In the expanded version of the polynomial from the above exercise, you'll see:

• \(1 - x + x^2\)
• \(+ x - x^2 + x^3\)
• \(+ x^2 - x^3 + x^4\)

Combine like terms by looking at the coefficients (numbers in front of the variables) and adding or subtracting them as needed:

• The constant term remains as is (1 in this case).
• For \(x\), you have \(-x + x\), which cancels out to 0.
• For \(x^2\), \(+x^2 - x^2 + x^2\) also results in 0.
• \(x^3\) terms also cancel each other out, \(+x^3 - x^3 = 0\).
• What remains is only \(1 + x^4\).

Combining like terms simplifies the polynomial, making it easier to read and understand.

Simplification Process

The simplification process in polynomial operations involves taking all the terms you've acquired from both distributing and combining like terms, and reducing them to their simplest form. Simplifying ensures that the polynomial is concise and error-free.

In the given exercise, after distributing and combining like terms, you're left with:
- \(1 + x^4\)
This is already in its simplest form because there are no more like terms to combine or further adjustments needed.

The outcome is clear and easy to interpret. Simplification not only makes the expression more manageable, it also minimizes potential errors in follow-up calculations or further algebraic manipulations. Remember, the ultimate goal of simplification is to have the cleanest expression possible, revealing the true nature of the polynomial.