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Magazine Chapter 5: Problem 62
Multiply each expression. $$ 4 y\left(y^{2}-8 y-4\right) $$
Short Answer
Expert verified
The product is \(4y^3 - 32y^2 - 16y\).
Step by step solution
01
Distribute the Monomial
The expression given is a monomial, \(4y\), multiplied by a trinomial, \(y^2 - 8y - 4\). To multiply these, distribute \(4y\) to each term in the trinomial. Start with multiplying \(4y\) by the first term \(y^2\).
02
Multiply Each Term
First, multiply \(4y \times y^2\): \[4y \times y^2 = 4y^3\]
03
Continue Distribution
Next, multiply \(4y \times (-8y)\): \[4y \times (-8y) = -32y^2\]
04
Complete the Distribution
Finally, multiply \(4y \times (-4)\): \[4y \times (-4) = -16y\]
05
Combine the Results
Combine all the terms you obtained from the distribution into one expression: \[4y^3 - 32y^2 - 16y\]
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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 simple algebraic expression that consists of only one term. It is composed of a coefficient and one or more variables raised to non-negative integer powers. A few examples of monomials are \(5x\), \(-3y^2\), and \(8a^3b^2\).
Monomials are the building blocks in algebra, with each term contributing to the overall expression.
Monomials are the building blocks in algebra, with each term contributing to the overall expression.
- A monomial like \(4y\) signifies that 4 is the coefficient and \(y\) is the variable.
- The power of the variable determines its degree. In \(4y\), the degree is 1.
- Monomials can be constants or values like \(x^3\) without a numerical coefficient (implicitly 1).
Trinomial
A trinomial is an algebraic expression that contains three terms. Just like monomials, trinomials are composed of coefficients and variables, but they have a combination of three distinct terms. For example, in the given expression \(y^2 - 8y - 4\), we can identify it as a trinomial.
- Trinomials typically follow the standard structure of \(ax^2 + bx + c\), though the variables and their exponents may vary.
- In \(y^2 - 8y - 4\), the terms are \(y^2\), \(-8y\), and \(-4\).
- Each term of a trinomial can represent different powers of the same variable, contributing to its complexity.
Distributive Property
The distributive property is a fundamental algebraic principle used to simplify expressions and is especially handy when multiplying a monomial by a polynomial, such as a trinomial. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.
- For example, when applying the distributive property to \(4y(y^2 - 8y - 4)\), you multiply \(4y\) by each term inside the parenthesis separately.
- This means: \(4y \times y^2, 4y \times (-8y),\) and \(4y \times (-4)\).
- The results are then combined: \(4y^3 - 32y^2 - 16y\).
Algebraic Expressions
Algebraic expressions are combinations of constants, variables, and arithmetic operations that result in a mathematical representation of a situation. They can range from simple monomials to complex polynomials that consist of multiple terms.
- They serve as a means to model real-world problems and are foundational in algebra.
- In \(4y(y^2 - 8y - 4)\), the entire expression is a product of simpler algebraic expressions: a monomial and a trinomial.
- Algebraic expressions can be manipulated using various algebraic principles, like combining like terms, the distributive property, and factoring.
