Problem 62 - Algebra 1: Concepts and Skills

Question

Find the product \(2 a^{2}\left(a^{2}-3 a+1\right)\) $$ \text { (A) } 2 a^{2}-6 a+2 $$ $$ \text { (B) } 2 a^{4}-6 a^{3}+2 a $$ $$ \text { (C) } 2 a^{2}-3 a^{3}+2 a^{2} $$ $$ \text { (D) } 2 a^{4}-6 a^{3}+2 a^{2} $$

Step-by-Step Solution

Answer

The product of the given expression corresponds to the choice \(D) 2a^4 - 6a^3 + 2a^2

Step 1: Distribute the \(2a^2\) across the Parentheses

Firstly, distribute the \(2a^2\) across each term within the parenthesis \(a^2 - 3a + 1\). This would look like: \(2a^2 * a^2 - 2a^2 * 3a + 2a^2 * 1\)

Step 2: Simplify the distributed terms

After distributing, simplify each term. The product of \(2a^2\) and \(a^2\) is \(2a^{2+2} = 2a^4\); the product of \(2a^2\) and \(3a\) is \(6a^{2+1} = 6a^3\); and the product of \(2a^2\) and 1 is simply \(2a^2\). The result will be: \(2a^4 - 6a^3 + 2a^2\)

Step 3: Compare with Given Options

Compare this simplified expression with the given options to determine the correct choice. From the four options, it is clear the result matches with the last option D: \(2a^4 - 6a^3 + 2a^2\)

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle used to multiply a single term by a group of terms within a parenthesis. It simplifies expressions and is vital for working with polynomials. In the exercise given, you start by distributing the term \(2a^2\) across each term within the parentheses \((a^2 - 3a + 1)\). This means:

• Multiply \(2a^2\) by \(a^2\)
• Multiply \(2a^2\) by \(-3a\)
• Multiply \(2a^2\) by \(1\)

Through the distributive property, each term inside the parentheses is effectively 'expanded' into individual products with the term outside (here \(2a^2\)). This process creates an expression that can be further simplified by combining like terms. Understanding this property is crucial in algebra because it allows for breaking down complex expressions into more manageable parts.

Monomials

Monomials are algebraic expressions that consist of only one term. They can include numbers, variables, and exponents. In our example, \(2a^2\) is a monomial. It's formed by a coefficient (which is \(2\) in this instance) and a variable with an exponent (\(a^2\)).

When multiplying monomials, such as \(2a^2\) with each term inside the parentheses \((a^2 - 3a + 1)\), it involves applying the laws of exponents to multiply the coefficients and the variables separately.

• First, the coefficients are multiplied to provide the new coefficient of the resulting monomial.
• Secondly, the exponents of the same base (in this case, the base \(a\)) are added together to get the new exponent.

This understanding helps in the simplification process and tackling more complex polynomial expressions.

Exponents

Exponents are components of algebraic expressions that denote how many times a number or variable is multiplied by itself. For instance, in the term \(a^2\), the exponent is 2, signifying that \(a\) is multiplied by itself once. In the problem at hand, understanding how exponents operate is crucial.

When we multiply terms with exponents, like \(2a^2\) by another \(a^2\), the exponents are added due to the law of exponents. This law states that when multiplying similar bases, we should add their exponents: \(a^2 \cdot a^2 = a^{2+2} = a^4\).

• This means the exponents produce a new power of the variable.
• The base remains unchanged unless multiplied by a different base.

Exponents are central in simplifying polynomial expressions, helping us to manage powers of variables and solve algebraic equations precisely.