Problem 29 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ \left(8 x^{3}+3\right)\left(x^{2}-5\right) $$

Step-by-Step Solution

Answer

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

Step 1: Distribute the Terms

First, distribute each term of the first polynomial \((8x^3+3)\) onto the terms of the second polynomial \((x^2-5)\). The result can be calculated as \(8x^3 \cdot x^2 + 8x^3 \cdot (-5) + 3 \cdot x^2 + 3 \cdot (-5)\).

Step 2: Simplify the Terms

Now, we have four multiplication expressions. Simplify each expression to get \(8x^5 - 40x^3 + 3x^2 - 15\). You calculate \(8x^3 \cdot x^2 = 8x^5\), \(8x^3 \cdot (-5) = -40x^3\), \(3 \cdot x^2 = 3x^2\), and \(3 \cdot (-5) = -15\).

Step 3: Combine like terms

In this case, there are no like terms to combine. Hence, the final result remains as \(8x^5 - 40x^3 + 3x^2 - 15\) after the simplification.

Key Concepts

Distributive Property

The distributive property plays a critical role when multiplying polynomials. It states that multiplying a sum by a number is equivalent to multiplying each addend by the number and then summing the products. For instance, if you have the expression \(a(b+c)\), using the distributive property, you would get \(ab + ac\).

When applying this to polynomial multiplication, such as \(8x^3+3)(x^2-5)\), you distribute each term of the first polynomial over each term of the second polynomial. The distributive step looks like this: \(8x^3 \times x^2 + 8x^3 \times (-5) + 3 \times x^2 + 3 \times (-5)\). This step-by-step unfolding ensures that all terms are accounted for in the subsequent multiplication, laying the groundwork for arriving at a simplified expression.

Simplify Expressions

After distributing the terms, the next step is to simplify the expression by performing the indicated multiplications. Simplification turns expanded polynomials into a more manageable form and involves reducing the expression to as few terms as possible.

In our example, we simplify the distributed terms to obtain \(8x^5 - 40x^3 + 3x^2 - 15\). During this phase, it's key to apply the laws of exponents correctly. When multiplying like bases, we add the exponents, for instance, \(x^3 \times x^2 = x^{3+2} = x^5\). Simplification is all about clarity and precision—make sure that each multiplication is carried out methodically, and accuracy will follow suit.

Combining Like Terms

The final step in polynomial multiplication involves combining like terms. Like terms are terms that have the same variable raised to the same power. They are the terms you can combine to make the expression even simpler.

In some cases, such as with our example \(8x^5 - 40x^3 + 3x^2 - 15\), there are no like terms to combine. However, if we had a situation where there were multiple terms with the same powers, then we would add (or subtract) their coefficients. For instance, if we had \(8x^2 + 3x^2\), we could combine them to get \(11x^2\) because both terms have the variable \(x\) raised to the second power. Understanding how to combine like terms is essential for keeping expressions as succinct as possible and for further algebraic manipulations.