Problem 30 - College Algebra

Question

Find each product. $$\left(8 x^{3}+3\right)\left(x^{2}-5\right)$$

Step-by-Step Solution

Answer

The product of the binomial \((8 x^{3}+3)\) and \((x^{2}-5)\) is \(8x^5 - 40x^3 + 3x^2 - 15\).

Step 1: Distribute the First Term of the First Parenthesis

The first term in the first parenthesis is \(8x^3\). Multiply this with each term in the second parenthesis which are \(x^2\) and -5 respectively. \n\n\(8x^3 \cdot x^2 = 8x^5\)\n\n\(8x^3 \cdot -5 = -40x^3\)

Step 2: Distribute the Second Term of the First Parenthesis

The second term in the first parenthesis is 3. Multiply this with each term in the second parenthesis, which are \(x^2\) and -5 respectively. \n\n\(3 \cdot x^2 = 3x^2\)\n\n\(3 \cdot -5 = -15\)

Step 3: Combine the Results

Combine the results obtained in step 1 and step 2 to get the final answer.\n\n\(8x^5 - 40x^3 + 3x^2 - 15\)

Key Concepts

Distributive Property

When multiplying polynomials such as \((8x^3 + 3)(x^2 - 5)\), one of the most important tools we use is the distributive property. This property allows us to multiply each term in the first polynomial by every term in the second polynomial.
The process involves distributing, or spreading out, the terms of one polynomial across the terms of another.
For example, in our exercise, for the expression \(8x^3\), we multiply it with each term in \(x^2 - 5\). This results in two products: \(8x^3 \cdot x^2\) which is \(8x^5\) and \(8x^3 \cdot (-5)\) which becomes \(-40x^3\).

• First Term Distribution: Multiply the first term of the first expression by each term of the second expression.
• Second Term Distribution: Repeat the same process for the second term of the first expression.

Using the distributive property effectively requires practice but is essential for simplifying or expanding algebraic expressions.

Exponents

Exponents play a critical role when multiplying algebraic expressions, especially when dealing with polynomials. They indicate how many times a number, known as the base, is multiplied by itself.
For example, in the exercise \(8x^5\), the number 8 is multiplied by \(x\) raised to the power of 5. This means \(x\) is multiplied by itself five times: \(x \cdot x \cdot x \cdot x \cdot x\).
When multiplying expressions with the same base, we add their exponents. So, for expression \(8x^3 \cdot x^2\), we add 3 and 2 to get 5, resulting in \(8x^5\). Understanding this rule will simplify many algebraic manipulations.

• The exponent denotes the number of times to multiply the base by itself.
• When multiplying like bases, add the exponents together.

Mastering exponents can make solving polynomial multiplications more manageable and less daunting.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators (like +, -, *). They can range from simple (like \(x + 3\)) to complex (like our polynomial \((8x^3 + 3)(x^2 - 5)\)).
To become proficient in manipulating algebraic expressions, it is crucial to identify the parts involved:

• Terms: Parts of the expression separated by + or - signs. For example, in \(8x^3 + 3\), \(8x^3\) and 3 are terms.
• Coefficient: The numerical part of a term that is multiplied by the variable, such as 8 in \(8x^3\).
• Variable: A symbol used to represent a number, often \(x, y, z\).

By understanding these components, students can effectively carry out operations such as addition, subtraction, and multiplication on algebraic expressions, leading to solutions for various mathematical problems.