Question

Add, subtract, or multiply, as indicated. Express your answer as a single polynomial in standard form. $$ 5 x^{3}(3 x-4) $$

Short answer

15x^4 - 20x^3

Step-by-step solution

Distribute the Polynomial

Multiply each term inside the parentheses by the term outside the parentheses. This involves using the distributive property: a(b + c) = ab + acApplying this to the given problem: 5x^3 (3x - 4) Multiply 5x^3 by 3x, and then by -4.

Perform the Multiplication

First, multiply 5x^3 by 3x: 5x^3 * 3x = 15x^4Next, multiply 5x^3 by -4: 5x^3 * -4 = -20x^3

Combine the Results

Combine the individual products to form the final polynomial expression: 15x^4 - 20x^3

Key concepts

Distributive Property

The distributive property is a fundamental concept in algebra. It helps you simplify expressions by distributing a multiplier over terms inside parentheses. The general form of the distributive property is: \( a(b + c) = ab + ac \).

When working with polynomials, this property allows you to break down complex multiplication problems into simpler steps. For example, in the exercise \( 5x^3 (3x - 4) \), you need to multiply \( 5x^3 \) by each term inside the parentheses.

Breaking it down, you multiply \( 5x^3 \) by \( 3x \), and then multiply \( 5x^3 \) by \( -4 \). These individual products are then combined to form the final expression.

Multiplication of Polynomials

Multiplying polynomials may seem complex, but it becomes simpler when you use rules like the distributive property. In the example given, we multiply two terms: \( 5x^3 \) and \( 3x - 4 \).

First, we multiply \( 5x^3 \) by \( 3x \):
\( 5x^3 \times 3x = 15x^4 \).

Here, you multiply the coefficients (5 and 3) and add the exponents of \( x \) (3 and 1). This gives \( 15x^4 \).
Next, we multiply \( 5x^3 \) by \( -4 \):
\( 5x^3 \times -4 = -20x^3 \).

Just be careful to apply the correct signs and combine the terms correctly to avoid mistakes.

Combining Like Terms

Combining like terms is essential in simplifying polynomial expressions. Like terms are terms that have the same variables raised to the same power. For example, \( 15x^4 \) and \( -20x^3 \) are unlike terms because the exponents of \( x \) are different (4 and 3).

To combine them, you simply add or subtract their coefficients. However, in our problem, we already have simplified like terms from our multiplication steps.

Finally, we write the combined terms together, resulting in the simplified polynomial.

Standard Form of a Polynomial

The standard form of a polynomial is an organized way to write polynomials. It involves arranging the terms in descending order of their degree (the highest exponent of the variable \( x \)).

For example, after multiplying and combining like terms in the exercise \( 5x^3 (3x - 4) \), we get \( 15x^4 - 20x^3 \). Here, \( 15x^4 \) comes first because it has the highest power (4), followed by \( -20x^3 \).

By arranging polynomials in standard form, you make them easier to read and understand. This form also helps in comparing polynomials and performing operations like addition and subtraction more efficiently.