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Chapter 5: Problem 35

Find each product of the monomial and the polynomial. $$4 x(x+3)$$

Short Answer

Expert verified
The product of the monomial and the polynomial \(4x(x+3)\) is \(4x^2 + 12x\).

Step by step solution

01

Understanding the Problem

In the given expression \(4x(x+3)\), the coefficients and variables need to be multiplied according to the distributive property, also called the distributive law of multiplication over addition.
02

Apply the Distributive Property

Next, multiply the monomial \(4x\) with each term of the binomial \((x+3)\). The product of \(4x\) and \(x\) is \(4x^2\). The product of \(4x\) and \(3\) is \(12x\). So the expression \(4x(x+3)\) simplifies to \(4x^2 + 12x\).
03

Summarize the Solution

The final solution to the original expression \(4x(x+3)\) is the simplified product \(4x^2 + 12x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Monomial
A monomial is a single mathematical expression that consists of a coefficient, which is a number, and one or more variables raised to whole number powers.
For example, in the expression \(4x\), "4" is the coefficient, and "x" is the variable. Together, they form a monomial because they represent a single term.
Monomials do not involve addition or subtraction within the term itself.
  • \(7y\) is a monomial 鈥� it has one term.
  • \(-3x^2\) is a monomial 鈥� it consists of a single variable raised to a power.
  • Constant numbers like 5 are also considered monomials because they can be thought of as \(5x^0\).
Understanding monomials is fundamental because they are building blocks for more complex algebraic expressions.
Polynomial
A polynomial is an algebraic expression that consists of one or more monomials separated by addition or subtraction. Each monomial in a polynomial is termed as a "term" of the polynomial.
For example, in the polynomial \(x + 3\), we have two terms: \(x\) and \(3\).
Polynomials can have varying numbers of terms and degrees:
  • A polynomial with one term is a monomial, like \(5x^2\).
  • A polynomial with two terms is a binomial, such as \(x + 3\).
  • A polynomial with three terms is a trinomial, like \(a^2 + 2a + 1\).
The degree of a polynomial is determined by the largest exponent of the variable in the polynomial.
Polynomials are essential in algebra because they can describe a broad range of relationships and they play a crucial role in equations and function modeling.
Multiplication of Algebraic Expressions
Multiplying algebraic expressions involves distributing each term across the other expression, known as the distributive property. This property says you multiply a single term within a bracket expression separately by each term inside the bracket.
For instance, consider the expression \(4x(x + 3)\):
  • First, distribute \(4x\) over the first term \(x\) to get \(4x \times x = 4x^2\).
  • Next, distribute \(4x\) over the second term \(3\) to obtain \(4x \times 3 = 12x\).
These individual products are then combined to form a new expression \(4x^2 + 12x\).
This approach of using the distributive property simplifies complex multiplications and is crucial for finding products in algebra.
Practice with different types of expressions will help in mastering this technique.

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Most popular questions from this chapter

Simplify: \((-10)(-7) \div(1-8)\)

determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. I used two points and a checkpoint to graph \(y=x^{2}-4\)

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