Problem 35 - Beginning and Intermediate Algebra

Question

Perform the indicated operations and simplify. \(\left(a^{2}-a+3\right)\left(a^{2}+4 a-2\right)\)

Step-by-Step Solution

Answer

The simplified expression for the given operation is: \(a^{4} + 3a^{3} - 3a^{2} + 14a - 6\).

Step 1: Write down the given expression

First, rewrite the expression: \(\left(a^{2}-a+3\right)\left(a^{2}+4 a-2\right)\)

Step 2: Apply the distributive property

Now, multiply each term of the first polynomial by each term of the second polynomial: \(a^{2}(a^{2})+a^{2}(4a)+a^{2}(-2)-a(a^{2})-a(4a)-a(-2)+3(a^{2})+3(4a)+3(-2)\)

Step 3: Perform the multiplications

Calculate the product for each pair of terms: \(a^{4} + 4a^{3} - 2a^{2} - a^{3} - 4a^{2} + 2a + 3a^{2} + 12a - 6\)

Step 4: Combine like terms

Add or subtract the coefficients of the like terms to simplify the expression: \(a^{4} + (4a^{3} - a^{3}) + (-2a^{2} - 4a^{2} + 3a^{2}) + (2a + 12a) - 6\)

Step 5: Simplify the coefficients

Perform the arithmetic operations on the coefficients of the like terms and simplify the expression: \(a^{4} + 3a^{3} - 3a^{2} + 14a - 6\) The simplified expression for the given exercise is: \(a^{4} + 3a^{3} - 3a^{2} + 14a - 6\)

Key Concepts

Distributive Property

When working with polynomial multiplication, the distributive property is one of the most essential tools. Think of it as a way to "distribute" each term of one polynomial across all the terms of another polynomial.
For instance, in our expression \((a^2 - a + 3)(a^2 + 4a - 2)\)the distributive property tells us to multiply each term in the first parentheses by each term in the second parentheses.

It starts with:

• \(a^2 \times a^2\)
• \(a^2 \times 4a\)
• \(a^2 \times (-2)\)
• \(-a \times a^2\)
• \(-a \times 4a\)
• \(-a \times (-2)\)
• \(3 \times a^2\)
• \(3 \times 4a\)
• \(3 \times (-2)\)

By multiplying out these pairs, you can then combine them in the next steps. This procedure ensures that each term gets properly accounted for, forming the basis for further simplification.

Combine Like Terms

After utilizing the distributive property, you'll end up with several expressions that may share common variables to the same power. These expressions are known as 'like terms'. 'Combining like terms' means adding or subtracting their coefficients.
This step helps in organizing the polynomial into a simpler form.

For example, from the expanded polynomial:

• \(a^4 + 4a^3 - a^3 - 2a^2 - 4a^2 + 3a^2 + 2a + 12a - 6\)

We can combine similar components:

• \(4a^3 - a^3 = 3a^3\)
• \(-2a^2 - 4a^2 + 3a^2 = -3a^2\)
• \(2a + 12a = 14a\)

This step is crucial in polynomial simplification as it reduces clutter, making the final expression easier to interpret and use.

Simplification of Expressions

Finally, once you've combined the like terms, the task left is simplifying the overall polynomial expression. Simplification involves performing basic arithmetic operations to achieve the simplest form possible.
The goal is an elegant and readable polynomial.

In our case:

• We're left with \(a^4 + 3a^3 - 3a^2 + 14a - 6\)

Each term has been simplified to its lowest form, with no further operations possible. Remember, simplification doesn't change the roots or the behavior of the polynomial; it simply presents it in a clean, straightforward manner.
This practice is useful not only in making a problem more manageable but also in helping you recognize properties of the polynomial, like degree and leading coefficient, at a glance.