Problem 24 - Algebra 1: Concepts and Skills

Question

$$ (-y)\left(6 y^{2}+5 y\right) $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(-6y^{3} - 5y^{2}\).

Step 1: Multiplication

Multiply \(-y\) with the first term inside the parenthesis, which is \(6y^{2}\). The multiplication of \(-y\) and \(6y^{2}\) results in \(-6y^{3}\).

Step 2: Multiplication Continued

Next, multiply \(-y\) with the remaining term inside the parenthesis, which is \(5y\). The multiplication of \(-y\) and \(5y\) yields \(-5y^{2}\).

Step 3: Combine

Combine both results from Step 1 and Step 2 to form a single expression. Given that both results are on the same level of the expression, this results in \(-6y^{3} - 5y^{2}\).

Key Concepts

Multiplying Polynomials

Understanding how to multiply polynomials is a vital skill in algebra. It involves combining two or more algebraic expressions by distributing each term of the first polynomial to every term of the second. Let's consider an example from the exercise: (-y)(6y^2 + 5y).
The process begins by taking the first term outside the parenthesis (-y), and multiplying it by each term inside the parenthesis (6y^2 and 5y). Distributive property states that a(b + c) = ab + ac. Following this rule, we first multiply -y by 6y^2 to get -6y^3 and then -y by 5y to obtain -5y^2. The final expression, after combining these products, is -6y^3 - 5y^2. Simplifying our expressions in this way allows us to turn a polynomial multiplication into an orderly single polynomial result. Pay attention to signs and ensure you maintain proper alignment of terms to facilitate easier addition and subtraction after multiplication.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. In our exercise, we encountered the algebraic expression 6y^2 + 5y
which is a binomial, because it consists of two terms. When multiplying an algebraic expression by a monomial (like -y), you must apply the distributive property, which is a cornerstone of algebra. By distributing properly, it's guaranteed that every term is multiplied, preserving the integrity of the mathematical operations. The exercise presents a concise example of this process. A common mistake occurs when students either miss multiplying all terms or incorrectly apply the distributive law, such as forgetting to multiply the coefficients (numerical values) or powers of variables correctly. Clarifying how to handle these components assures proper simplification and manipulation of algebraic expressions.

Negative Exponents

Negative exponents refer to the concept of division in another guise. They mean that a number or variable is divided, rather than multiplied, a certain number of times. In the realm of algebra, negative exponents invert the base (the number or variable they're attached to) and transform them into their reciprocal. Although in our exercise we don't directly deal with negative exponents, understanding them is important when simplifying algebraic expressions that result from polynomial multiplication, especially when division involves variables with exponents. It's crucial to note that a negative exponent does not make the whole term negative but rather indicates the reciprocal, so for instance x^{-1} = 1/x.
Remember, working with negative exponents often requires careful attention to where the base is moved, whether to the numerator or denominator, in order to transform the exponent into a positive one.