Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

\(9t^4 - 6t^3 + 7t^2 - 2t + 1\)

Step 1: Understand the Expression

We are given the expression \((3t^2 - t + 1)^2\). This is a binomial expression raised to a power, specifically squared. To simplify this expression, we need to expand it.

Step 2: Write Using the Formula

To square a binomial, we use the formula \((a - b + c)^2 = (a - b + c)(a - b + c)\). In our case, we apply this to \(3t^2 - t + 1\).

Step 3: Expand the Expression

Use the distributive property (also known as the FOIL method) to expand:\[(3t^2 - t + 1)(3t^2 - t + 1).\] First, distribute \(3t^2\): \[3t^2(3t^2 - t + 1) = 9t^4 - 3t^3 + 3t^2.\]

Step 4: Distribute the Middle Term

Now, distribute \(-t\): \[-t(3t^2 - t + 1) = -3t^3 + t^2 - t.\]

Step 5: Distribute the Last Term

Finally, distribute \(1\): \[1(3t^2 - t + 1) = 3t^2 - t + 1.\]

Step 6: Combine Like Terms

Gather all the terms from the expansion:\[9t^4 - 3t^3 + 3t^2 - 3t^3 + t^2 - t + 3t^2 - t + 1.\]Combine like terms:\[9t^4 - 6t^3 + 7t^2 - 2t + 1.\]

Step 7: Simplify the Expression

After combining like terms, the simplified expression is:\[9t^4 - 6t^3 + 7t^2 - 2t + 1.\]

Key Concepts

Binomial Theorem

The Binomial Theorem is a helpful tool for expanding expressions that are raised to a power. In our case, the expression is a trinomial squared, which still aligns with the concepts behind the Binomial Theorem.
The theorem provides a method to expand expressions of the form \((a + b)^n\). However, in our exercise, we have \((3t^2 - t + 1)^2\). While it is not a simple binomial, the same idea can be applied: multiplying the trinomial by itself.
For a trinomial, each term needs to be multiplied separately using the distributive property, ensuring we apply the squared to every pair of terms.

Distributive Property

The Distributive Property is fundamental in algebra, especially when expanding expressions like \[ (3t^2 - t + 1)(3t^2 - t + 1) \]. It states that a term multiplied across an addition (or subtraction) can be distributed to each part within the parentheses.
It's sometimes remembered by the mnemonic FOIL, which stands for First, Outer, Inner, and Last when dealing with two binomials.
Particularly in this exercise:

• First, multiply \(3t^2\) with every term in the second polynomial.
• Second, multiply \(-t\) with every term.
• Lastly, distribute the constant \(1\) throughout.

Each step helps set the stage for combining and simplifying later.

Polynomial Simplification

Polynomial Simplification involves combining like terms to create a cleaner and more streamlined expression. After applying the distributive property in our given exercise, we've got a lot of terms to combine.
Combining terms means adding or subtracting coefficients of terms with the same variables and exponents.
For instance:

• The terms \(-3t^3\) add up to result in \(-6t^3\),
• and terms involving \(t^2\) combine to form \(7t^2\).

Simplifying is critical as it provides us with a concise polynomial from our expanded form: \[ 9t^4 - 6t^3 + 7t^2 - 2t + 1 \].

Algebraic Expressions

Algebraic expressions incorporate numbers, variables, and sometimes operations (like additions and multiplications), as seen in our problem with \(3t^2 - t + 1\). Understanding these helps in manipulating and transforming expressions correctly during operations such as expansion and simplification.
In algebra, you often have to identify parts of an expression, such as:

• "Terms," like \(3t^2, -t, 1\), each of which can be constant or have variables.
• "Coefficients," which are numbers multiplying the variables, like the \(3\) in \(3t^2\).
• "Variables," typically represented by letters like \(t\), which stand in for unknown values.

These components are the building blocks of more complex formulas we tackle in algebra.