Problem 49 - Algebra A Combined Function

Question

Multiply. \((2 a-3)\left(5 a^{2}-6 a+4\right)\)

Step-by-Step Solution

Answer

The product is \(10a^3 - 27a^2 + 26a - 12\).

Step 1: Distribute the First Term

Multiply the first term of the binomial \(2a\) by each term inside the trinomial \(5a^2 - 6a + 4\), using the distributive property. 1. \(2a \times 5a^2 = 10a^3\)2. \(2a \times (-6a) = -12a^2\)3. \(2a \times 4 = 8a\)

Step 2: Distribute the Second Term

Next, multiply the second term of the binomial \(-3\) by each term inside the trinomial.1. \(-3 \times 5a^2 = -15a^2\)2. \(-3 \times (-6a) = 18a\)3. \(-3 \times 4 = -12\)

Step 3: Combine Like Terms

Combine the results from Steps 1 and 2, and add the expressions together by combining like terms:\(10a^3 -12a^2 + 8a - 15a^2 + 18a - 12\)1. Combine \(-12a^2\) and \(-15a^2\): \(-12a^2 - 15a^2 = -27a^2\)2. Combine \(8a\) and \(18a\): \(8a + 18a = 26a\)3. The final expression is: \(10a^3 - 27a^2 + 26a - 12\)

Key Concepts

Distributive Property

The distributive property is a crucial algebraic rule that allows us to simplify and solve expressions or equations. It involves distributing, or multiplying, a single term by each term in a parenthesis. In our exercise, we use this property to multiply terms from a binomial with terms from a trinomial. This can be visualized as spreading the multiplication over all terms included in parentheses.

• First, you multiply the first term of the binomial by each term in the trinomial.
• Then, repeat the process with the second term of the binomial.

This property is foundational for polynomial multiplication, enabling the breakdown of complex expressions into simpler parts. By applying the distributive property, we meticulously handle smaller calculations which are easier to manage and combine later.

Combining Like Terms

Combining like terms is the process of simplifying an expression by merging terms that have the same variable parts. The key here is to identify terms that share both the variable and its exponent.

• Terms such as \(-12a^2\) and \(-15a^2\) are like terms because they both contain the variable \(a^2\).
• Similarly, \(8a\) and \(18a\) are like terms because they share the \(a\) variable.

By adding or subtracting their coefficients, we simplify the expression to make it more understandable and manageable. In our example, after performing the multiplication, you need to combine these like terms to arrive at the final straightforward polynomial expression. This is a critical step in reducing complex algebraic expressions into their simplest form.

Binomial

A binomial is a polynomial that contains exactly two terms. This concept is part of polynomial algebra where terms are connected by a plus or minus sign. In the context of our exercise:

• The binomial is \((2a-3)\), which comprises two distinct terms: \(2a\) and \(-3\).
• These terms are coefficients and variables or standalone constants.

Binomials are the simplest forms of polynomials beyond monomials (single-term expressions), and they often serve as the building blocks for more complex algebraic operations. To effectively multiply binomials with other polynomials, recognizing and correctly using the distributive property is essential.

Trinomial

A trinomial is a type of polynomial that consists of three terms. These terms are combined with addition or subtraction operators, and they include coefficients and variables raised to specific powers. In the given exercise, the trinomial is \((5a^2 - 6a + 4)\).

• It consists of the different terms: \(5a^2\), \(-6a\), and \(4\).
• These terms represent parts of the polynomial expression that cannot be combined without further context, such as distribution and combining like terms.

Trinomials are broader than binomials and typically involve a variety of algebraic operations, including the multiplication of binomials. Understanding trinomials helps in extending basic algebraic principles to solve more intricate polynomial equations efficiently.