Problem 18 - Elementary Algebra

Question

Find the following products and simplify. $$ \left(3 a^{2}-1\right)\left(5 a^{2}+a\right) $$

Step-by-Step Solution

Answer

Answer: The simplified product of the given polynomials is \(15a^4 + 3a^3 - 5a^2 - a\).

Step 1: Distribute the first term of the first polynomial

Multiply the first term of the first polynomial, \(3a^2\), with each term in the second polynomial, which are \(5a^2\) and \(a\). $$ (3a^2)(5a^2) = 15a^4 $$ $$ (3a^2)(a) = 3a^3 $$ So far, we have: $$ 15a^4 + 3a^3 $$

Step 2: Distribute the second term of the first polynomial

Multiply the second term of the first polynomial, \(-1\), with each term in the second polynomial, which are \(5a^2\) and \(a\). $$ (-1)(5a^2) = -5a^2 $$ $$ (-1)(a) = -a $$ So far, we have: $$ -5a^2 -a $$

Step 3: Combine the results of Steps 1 and 2

Add the results from Step 1 and Step 2 as shown below: $$ (15a^4 + 3a^3) + (-5a^2 -a) $$ Combining the terms, we get: $$ 15a^4 + 3a^3 - 5a^2 - a $$

Step 4: Write the final answer

The simplified product of the given polynomials is: $$ 15a^4 + 3a^3 - 5a^2 - a $$

Key Concepts

Distributive Property

When multiplying polynomials, the distributive property is a powerful tool to help manage the multiplication process, especially when dealing with expressions that consist of multiple terms. The distributive property allows us to multiply a single term by every term in another expression. Essentially, it helps break down complex multiplication into simpler steps.
To apply the distributive property, you take each term from one polynomial and multiply it by each term in the second polynomial. In our example, \(3a^2 - 1\) and \(5a^2 + a\), for each term in the first polynomial (like \(3a^2 \) and \(-1\)), you distribute it over every term in the second polynomial (like \(5a^2 \) and \(a\)).
Steps include:

• Multiplying \(3a^2\) with \(5a^2\) and \(a\) gives you \(15a^4\) and \(3a^3\).
• Then distributing \(-1\) gives you \(-5a^2\) and \(-a\).

By distributing each term separately and then combining the results, we simplify the multiplication process, ensuring that all terms are accounted for.

Simplification

Simplification in the context of polynomial multiplication involves combining like terms to express the final result in the simplest form possible. Once you have multiplied each term across the polynomials using the distributive property, the next step is to tidy up the expression.
Combining like terms means adding or subtracting terms that contain the same variable raised to the same power. In a polynomial, terms with different powers of a variable (like \(a^4, a^3, a^2\)) are distinct, but if two terms are both say \(a^2\), they can be combined.
In our example, after distribution, the result was \(15a^4 + 3a^3 - 5a^2 - a\). Since there are no terms with the same degree, the expression is already simplified. Sometimes, you might need to rearrange and further reduce terms, but in this case, each term is separate and cannot be combined with others. This expression is as simplified as it can be.

Polynomial Expressions

Polynomial expressions are mathematical expressions that consist of variables raised to whole number powers and their coefficients. They can have one or multiple terms and can involve adding, subtracting, and multiplying these terms. Polynomials are expressed in the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where each \(a_i\) is a coefficient and \(x\) is a variable.
Understanding polynomials and their structures is essential when dealing with operations like addition, subtraction, and especially multiplication. For multiplication, recognizing the degree of polynomial (the highest power of the variable in the expression) can help anticipate the degree of the resulting polynomial after operations.
In our example, the expression \(3a^2 - 1\) is a polynomial of degree 2, and \(5a^2 + a\) is another polynomial of degree 2. When multiplied, the resulting polynomial, \(15a^4 + 3a^3 - 5a^2 - a\), is of degree 4. The degree of the final polynomial was determined by the term with the highest power, which here is \(a^4\), resulting from multiplying \(a^2 \) by itself.