Problem 104 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ (a-4)\left(a^{2}+a-5\right) $$

Step-by-Step Solution

Answer

Answer: The result of the multiplication is a^3 - 3a^2 - 9a + 20.

Step 1: Multiply each term in the binomial by each term in the trinomial

Using the distributive property, we start by multiplying \((a-4)\) by each term in the trinomial \((a^2+a-5)\): $$ (a-4)(a^2+a-5) = a(a^2+a-5) - 4(a^2+a-5) $$ Now, we perform the multiplications: $$ = a(a^3 + a^2 - 5a) - 4(a^2 + a - 5) $$

Step 2: Distribute each term in the binomial

Now, distribute each of the terms in the binomial to all the terms of the trinomial: $$ = (a^3 + a^2 - 5a) - (4a^2 + 4a - 20) $$

Step 3: Combine like terms

Finally, sum all of the like terms in the expression: $$ = a^3 + a^2 - 5a - 4a^2 - 4a + 20 $$ Combine the \(a^2\) terms and the \(a\) terms: $$ = a^3 +(1 - 4)a^2 +(-5 - 4)a + 20 $$ Simplify: $$ = a^3 - 3a^2 - 9a + 20 $$ The final solution is: $$ (a-4)(a^2+a-5) = a^3 - 3a^2 - 9a + 20 $$

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows for the multiplication of a single term by terms within parentheses. It's often phrased as, "distribute the outside term to everything inside the parentheses." This property is essential when multiplying polynomials, as it lays the groundwork for how each part of the equation interacts. For example, the expression

• (a-4)(a^2 + a - 5)

is expanded by applying the distributive property, resulting in two separate distributions:

• a(a^2 + a - 5)
• -4(a^2 + a - 5)

These multiplications ensure every term from the first polynomial is multiplied by every term of the second. After the distributions, the expression appears as:

• a^3 + a^2 - 5a
• - 4a^2 - 4a + 20

Using this property ensures accuracy in polynomial expansion and later, in simplifying the combined product.

Combining Like Terms

Once all terms have been multiplied through, the next step involves simplifying the expression by combining like terms. Like terms are terms in an algebraic expression that have the same variable raised to the same power. For instance, in the expression

• (a^3 + a^2 - 5a - 4a^2 - 4a + 20)

we group terms to identify those which can be combined:

• a^2 terms: a^2 and -4a^2
• a terms: -5a and -4a

Combining these involves simple addition or subtraction of their coefficients, while ensuring the power of the variables remains unchanged. So it becomes:

• a^3 + (1 - 4)a^2 + (-5 - 4)a + 20

This simplifies further to:

• a^3 - 3a^2 - 9a + 20

Combining like terms makes expressions simpler and more manageable, setting up for either further operations or the solution completion.

Binomial Expansion

Binomial expansion involves the process of expanding expressions raised to a power or multiplying expressions where terms are in groups of two. In simpler terms, it often means multiplying a binomial with another polynomial. The initial expression

• (a-4)(a^2 + a - 5)

is a typical application of binomial expansion. The objective is to multiply every term in the binomial with the trinomial's every term, leading to a fully expanded polynomial. After thoroughly expanding with the distributive property, as shown:

• a(a^2 + a - 5) yields a^3 + a^2 - 5a
• -4(a^2 + a - 5) yields -4a^2 - 4a + 20

These steps faithfully execute the method of binomial expansion, ensuring all components are combined systematically. It enhances understanding by showcasing how seemingly complex expressions break down logically through step-by-step multiplication and distribution. Binomial expansions are a foundational pillar for working with more intricate algebraic expressions.