Question

Find the product. \(11 x\left(-4 x^2+3 x+8\right)\)

Short answer

The product is \(-44x^3 + 33x^2 + 88x\).

Step-by-step solution

Identify The Terms

Identify the terms that need to be multiplied. We have \(11x\), \(-4x^2\), \(3x\), and \(8\).

Multiply the Terms Using Distributive Property

The distributive property of multiplication over addition states that we need to multiply the monomial with each term of the polynomial: \(11x*(-4x^2) = -44x^3\), \(11x*(3x) = 33x^2\), and \(11x*8 = 88x\)

Write the Final Solution

Combine all the terms to get the final product. Thus, the product of \(11x\) and \(-4x^2 + 3x + 8\) is \(-44x^3 + 33x^2 + 88x\)

Key concepts

Distributive Property

The distributive property is a fundamental concept in mathematics. It enables us to simplify expressions by distributing multiplication over addition or subtraction. This means if you have a term outside the parentheses, you multiply it by each term inside the parentheses. This property is incredibly useful when dealing with polynomials. Let's see it in action:

• When given an expression like \(11x(-4x^2 + 3x + 8)\), you'll apply the distributive property by multiplying \(11x\) with each term inside the parentheses.
• This results in \(11x \cdot (-4x^2)\), \(11x \cdot 3x\), and \(11x \cdot 8\).
• Each multiplication is performed separately, allowing the distribution to simplify into one combined polynomial.

Understanding how to use the distributive property helps in simplifying calculations and arriving at the correct solution in polynomial multiplication scenarios.

Monomial

A monomial is a single term consisting of a constant, a variable, or a combination that includes numbers and variables multiplied together. It can be simple, like \(5\), or complex, such as \(11x\), which is what we see in our problem.

• The monomial in our expression is \(11x\), consisting of the constant 11 and the variable \(x\).
• One of the critical operations we perform with monomials is multiplication, especially when they are part of an expression involving polynomials.
• When multiplying a monomial by a polynomial, like \(11x \times (-4x^2 + 3x + 8)\), each term in the polynomial is multiplied by the monomial separately.

Recognizing a monomial and knowing how to operate with it forms the basis for understanding more complex algebraic expressions.

Multiplying Polynomials

Multiplying polynomials involves distributing each term in one polynomial to every term in another polynomial. This process can seem daunting, but it follows a systematic approach. In the exercise at hand, we multiplied a monomial by a polynomial, but the same principles apply to polynomials of higher degrees.

• Take each term of the polynomial and multiply it by every term in the other polynomial, using the distributive property.
• Combine like terms if necessary. Like terms are those that have the same variable raised to the same power.
• For example, from our step-by-step solution, multiplying \(11x\) by each term of \(-4x^2 + 3x + 8\) gives the results: \(-44x^3\), \(33x^2\), and \(88x\).

By conducting the multiplication using the distributive property, the result is streamlined to produce a single polynomial expression. Mastering polynomial multiplication is key to tackling more advanced algebra topics.