Problem 17 - Algebra 2 and Trigonometry

Question

Perform the indicated operations and write the result in simplest form. \((y-1)\left(y^{2}-2 y+1\right)\)

Step-by-Step Solution

Answer

The result is \(y^3 - 3y^2 + 3y - 1\) in simplest form.

Step 1: Identify the Expression

We need to perform the operation on the expression \((y-1)(y^{2}-2y+1)\). This expression involves two parts: a binomial and a trinomial. We will proceed by expanding this using the distributive property.

Step 2: Distribute the Binomial

Apply the distributive property: each term in the binomial \((y-1)\) must be multiplied by each term in the trinomial \((y^2-2y+1)\).First, distribute the \(y\):- \(y imes y^2 = y^3\)- \(y imes -2y = -2y^2\)- \(y imes 1 = y\)Then, distribute the \(-1\):- \(-1 imes y^2 = -y^2\)- \(-1 imes -2y = 2y\)- \(-1 imes 1 = -1\)

Step 3: Combine Like Terms

Now, combine the like terms from the distributed expression. The expression from Step 2 expands to:\[y^3 - 2y^2 + y - y^2 + 2y - 1\]Combine the quadratic terms:\(-2y^2 - y^2 = -3y^2\).Combine the linear terms:\(y + 2y = 3y\).This gives the simplified expression:\[y^3 - 3y^2 + 3y - 1\]

Step 4: Write the Result in Simplest Form

After combining like terms, we've arrived at the simplest form of the expression:\[y^3 - 3y^2 + 3y - 1\].This expression is already in its simplest form, as there are no like terms left to combine.

Key Concepts

Binomial Multiplication

Binomial multiplication is a fundamental algebraic process. It involves multiplying two terms, commonly called binomials. In this exercise, our binomial is

• \((y-1)\)

The other part is a trinomial:

• \((y^2-2y+1)\)

To execute binomial multiplication, you apply each term in the binomial separately to every term in the trinomial. This systematic approach ensures that every possible product is accounted for. Remember:

• First, multiply each term of the binomial by the first term of the trinomial.
• Then, repeat for the second term.

By understanding this approach, you build a solid foundation for handling more complex polynomial expressions later.

Trinomial Expansion

Trinomial expansion occurs when a trinomial is expanded by distribution. A trinomial comprises three distinct terms, as seen in

• \(y^2-2y+1\)

In this case, you are expanding it by multiplying it with each term from another expression—in our example, a binomial. The key here is to apply the distributive property, which involves making sure every term in one set multiplies each term in the other set. The expansion reveals all component products, from which you can then begin simplifying. Successfully expanding a trinomial establishes a comprehensive view of the equation, preparing you to combine like terms effectively in the next step.

Distributive Property

The distributive property is a core principle used to simplify algebraic expressions. It states that a single term can be distributed across terms within parentheses, maintaining the equality of the expression. In practical terms, this means if you have an expression like

• \((a+b)(c+d)\)

You distribute each term from

• \((a+b)\)

into

• \((c+d)\)

This yields products like

• \(ac+ad+bc+bd\)

In the exercise, this property guided the multiplication of the terms in our binomial with those in the trinomial. Using the distributive property ensures precision in breaking down complex expressions into manageable pieces, ready for further simplification.

Combining Like Terms

Once you have expanded an expression through distribution, the goal is to combine like terms. Like terms share the same variable raised to the same power. In the context of our exercise, we had:

• Quadratic terms \(-2y^2\) and \(-y^2\)
• And linear terms \(y\) and \(2y\)

By identifying these, you can sum them up:

• \(-2y^2 - y^2 = -3y^2\)
• \(y + 2y = 3y\)

Combining like terms simplifies the expression significantly. This step is crucial because it reduces the expanded expression to its simplest form, making it clear and concise. Properly doing this ensures you fully understand the equation's layout and properties.