Question

$$ \text { perform the indicated operations and simplify. } $$ $$ \left(3 t^{2}-t+1\right)^{2} $$

Short answer

The simplified expression is \(9t^4 - 6t^3 + 7t^2 - 2t + 1\).

Step-by-step solution

Recognize the Formula

The given operation is the square of a binomial, \((3t^2 - t + 1)^2\). We can use the formula for squaring a trinomial: \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\). Here, \(a = 3t^2\), \(b = -t\), and \(c = 1\).

Compute Each Term

We will apply the formula:- **\(a^2\):** \((3t^2)^2 = 9t^4\).- **\(b^2\):** \((-t)^2 = t^2\).- **\(c^2\):** \((1)^2 = 1\).- **\(2ab\):** \(2\cdot 3t^2\cdot(-t) = -6t^3\).- **\(2bc\):** \(2\cdot(-t)\cdot1 = -2t\).- **\(2ca\):** \(2\cdot1\cdot3t^2 = 6t^2\).

Combine All Terms

Add all the calculated terms together to simplify:\(9t^4 + t^2 + 1 - 6t^3 - 2t + 6t^2\).

Simplify the Expression

Combine like terms in the expression:\(9t^4 - 6t^3 + (t^2 + 6t^2) - 2t + 1\) simplifies to:\(9t^4 - 6t^3 + 7t^2 - 2t + 1\).

Key concepts

Binomial Theorem

The Binomial Theorem is a powerful method used for expanding expressions that are raised to a power. It typically applies to binomials—expressions with two terms—enabling us to expand them without tedious multiplication. In the given exercise, we are actually dealing with a trinomial (three terms) rather than a simple binomial, so we have adjusted using a closely related approach, one that builds upon the principles of squaring a binomial.When you encounter a binomial like \((a + b)^n\), the binomial theorem outlines how each term in the expanded expression will appear and how its coefficients can be determined. Although our exercise involves a trinomial, the principle of breaking down the expansion into calculable parts remains consistent. This process means identifying each component expression's unique contribution to the final expanded result and applying multiplication rules appropriately.Breaking down a trinomial involves recognizing patterns or set structures, like in our modified trinomial formula: \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\). Using structured approaches like this simplifies calculations drastically, helping guide the process smoothly and accurately for successful operations.

Expression Simplification

Expression simplification revolves around the idea of transforming a complex expression into the simplest form without changing its value. Simplifying involves various algebraic operations including distributing terms, combining like terms, and reducing fractions if necessary.In our exercise, after expanding the squared trinomial, we gather all the resulting terms to form a complete expression like \(9t^4 + t^2 + 1 - 6t^3 - 2t + 6t^2\). The goal here is to condense this expression as much as possible, which involves careful organizing and aligning terms such that we reduce clutter.Transformation techniques like these are valuable in algebra to represent results concisely. They allow mathematicians to spot patterns quickly, facilitate computations, and unveil results that might not be evident amidst a jumble of terms. Simplification is not about changing the expression's value; it's all about making it more understandable and more usable for further mathematical operations.

Like Terms

Understanding like terms is fundamental in simplifying expressions, enabling you to consolidate similar components and handle them collectively. Like terms refer to terms in an algebraic expression that have identical variables raised to the same power, but they can have different coefficients.For example, in our expanded expression \(9t^4 - 6t^3 + 7t^2 - 2t + 1\), terms like \(t^2\) are considered to be like terms. So, when terms like \(t^2 + 6t^2\) appear in the process, you combine them to get \(7t^2\) as we've seen in the solution.Identifying and combining like terms lead to a more simplified, elegant expression. Recognizing these allows mathematicians and students alike to perform additions, subtractions, and comparisons more efficiently, cleaning up expressions and rendering them more readable. This practice is crucial because it helps uphold accuracy in calculations and reduces errors, promoting a more streamlined mathematical workflow.