Problem 16 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ (x+5)\left(x^{2}-5 x+25\right) $$

Step-by-Step Solution

Answer

The product of the given binomial and trinomial is \(x^3 + 125\).

Step 1: Distribute the Binomial into the Trinomial

Start by distributing \(x+5\) into all the terms of the trinomial \(x^2 - 5x + 25\). So, the equation will be \(x*(x^2) - x*(5x) + x*25 + 5*(x^2) - 5*5x + 5*25\).

Step 2: Simplify

Now, we simplify the equation to obtain \(x^3 - 5x^2 + 25x + 5x^2 - 25x + 125\). As we see in this equation, the \(25x\) and the \(-25x\) will cancel each other out.

Step 3: Combine like terms

Next, combine like terms together. So \(x^3 - 5x^2 + 5x^2 = x^3\). Hence, the equation simplifies to \(x^3 + 125\).

Key Concepts

Binomial Expansion

Binomial expansion involves expanding a polynomial expression that includes two terms. In simple terms, a binomial is like an expression consisting of two terms, for example, \((x + 5)\). When we talk about expanding a binomial, especially in cases where it's multiplied by a trinomial like in the given exercise, the goal is to distribute each term from the binomial across every term in the other polynomial.

• The binomial here is \(x + 5\).
• The trinomial is \(x^2 - 5x + 25\).

Binomial expansion helps in breaking down complex polynomial multiplications into simpler, manageable pieces. Each term from the binomial gets multiplied by every term of the trinomial. For instance, \(x\) from \(x + 5\) is multiplied with every individual term in the trinomial. Similarly, \(5\) from \(x + 5\) is also multiplied with each term of the trinomial. This approach ensures a complete expansion, making sure all parts of both polynomials are multiplied together.

Distributive Property

The distributive property is a fundamental arithmetic idea used to multiply a single term with terms inside parentheses. In this exercise, it allows us to break down the multiplication of a binomial by a trinomial into simple products that are easy to handle. Here's a closer look at how it works:

• The distributive property states that \(a(b + c) = ab + ac\).
• In our exercise, we use this idea twice: once for \(x\) and once for \(5\) from the binomial \((x + 5)\).

By applying the distributive property, each term in the binomial multiplies everything in the trinomial. So \(x\) is multiplied by \(x^2 - 5x + 25\), and the result is added to what's obtained when \(5\) is multiplied by \(x^2 - 5x + 25\).This step by step application allows us to simplify very complex polynomial expressions. Make sure each term is correctly distributed, ensuring accurate solutions.

Combine Like Terms

Once the multiplication using the distributive property is complete, we end up with several terms. The next important step is to combine like terms to simplify our expression down into its simplest form. Like terms in algebra are terms that have the same variables raised to the same power. For example, \(-5x^2\) and \(5x^2\) can be combined because they are like terms.
By combining the like terms, the equation from our original distribution results in:

• We have \(x^3\)
• Observe that \(-5x^2 + 5x^2 = 0\): These terms cancel each other
• Similarly, \(25x - 25x = 0\): Again, these terms cancel each other out

After combining and simplifying, we are left with \(x^3 + 125\). This illustrates the power of combining like terms, which removes unnecessary parts of the expression, focusing only on the essential components to achieve a simple and elegant polynomial expression.