Problem 70 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 4 x\left(3 x^{2}-6 x+10\right) $$

Step-by-Step Solution

Answer

Question: Multiply the monomial 4x with the polynomial 3x^2-6x+10 and simplify the expression. Answer: 12x^3 - 24x^2 + 40x

Step 1: Multiply the monomial with each term of the polynomial

To do this, we use the distributive property of multiplication over addition, which means that we multiply the monomial with each term of the polynomial separately. So, we have: $$ 4x(3x^2) - 4x(6x) + 4x(10) $$

Step 2: Simplify each product

Now, we simplify each multiplication: $$ (4\times3)x^{1+2} - (4\times6)x^{1+1} + (4\times10)x^1 $$ $$ 12x^3 - 24x^2 + 40x $$

Step 3: Combine like terms if necessary

In this problem, there are no like terms to combine since all the terms have different exponents. So, our final simplified expression is: $$ 12x^3 - 24x^2 + 40x $$

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental principle of algebra that allows us to simplify expressions and solve equations efficiently. It states that a term outside the parentheses can be distributed, or multiplied, by each term inside the parentheses. This property is particularly useful when dealing with expressions that involve both monomials and polynomials.
Here's the basic idea:

• If you have an expression like \( a(b + c) \), the distributive property allows you to rewrite this as \( ab + ac \).
• This means that the single term \( a \) is multiplied by each term inside the brackets separately, then added together.

In our original exercise, we applied the distributive property to multiply the monomial \( 4x \) with each term of the polynomial \( 3x^2 - 6x + 10 \). This is written out as:
\[4x(3x^2) - 4x(6x) + 4x(10)\]
By applying the distributive property, we simplify the expression step-by-step.

Identifying Monomials and Polynomials

In algebra, understanding the difference between monomials and polynomials is crucial for solving various kinds of problems, including multiplication tasks.
A **monomial** is a single term expression consisting of a constant multiplied by one or more variables raised to a power. Examples include \( 2x \), \( 5y^3 \), or even just a constant like \( 7 \). It has no addition or subtraction among its terms.
On the other hand, a **polynomial** is a sum of multiple monomials, which means it could have several terms. For instance, \( 3x^2 - 6x + 10 \) is a polynomial with three terms.
During polynomial multiplication, each monomial term in the polynomial is treated separately using operations like the distributive property. In our problem, the monomial \( 4x \) is multiplied individually with each term of the polynomial \( 3x^2 - 6x + 10 \). This step helps in managing complex polynomials and simplifies calculation as demonstrated in the exercise.

Combining Like Terms

Combining like terms is a process used to simplify algebraic expressions. This concept is important because it allows you to make expressions more manageable and easy to work with. Like terms are terms in an expression that have the same variable raised to the same power. They only differ in their coefficient. For example:

• \( 3x \) and \( 7x \) are like terms because they both have the variable \( x \) raised to the first power.
• However, \( 3x^2 \) and \( 3x \) are not like terms because the powers of \( x \) are different.

The process of combining like terms involves adding or subtracting the coefficients of terms that have the same variable component. In our exercise, after multiplying, we are left with three distinct terms: \( 12x^3 \), \( -24x^2 \), and \( 40x \), which cannot be further combined as they have different exponents.
This step is essential for ensuring the expression is in its simplest form.