Problem 23 - Elementary Algebra

Question

Multiply. $$ -13(9 y 2-3 y+27) $$

Step-by-Step Solution

Answer

-117y^2 + 39y - 351

Step 1: Distribute the First Term

Distribute \(-13\) to each term inside the parentheses. First, distribute to \(9y^2\): \(-13 imes 9y^2 = -117y^2\).

Step 2: Distribute the Second Term

Next, distribute \(-13\) to \(-3y\): \(-13 imes (-3y) = 39y\).

Step 3: Distribute the Third Term

Finally, distribute \(-13\) to \(27\): \(-13 imes 27 = -351\).

Step 4: Write the Final Expression

Combine all the results from the distribution:\(-117y^2 + 39y - 351\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps when multiplying a single term by a polynomial. It allows us to distribute the multiplication over addition or subtraction inside parentheses, and it looks like this:

• If you have an expression like \(-a(b + c)\), you will multiply \(-a\) by each term inside the parenthesis, resulting in \(-ab - ac\).
• This property is useful because it breaks down complex expressions into a series of simpler operations, making calculations more manageable.

In the given exercise, \(-13(9y^2 - 3y + 27)\), the distributive property is applied to multiply \(-13\) with each term inside the parentheses:

• First, \(-13 \times 9y^2 = -117y^2\).
• Then, \(-13 \times (-3y) = 39y\).
• And finally, \(-13 \times 27 = -351\).

These steps showcase the distributive property, ensuring each term is handled accurately.

Polynomial Expressions

Polynomial expressions consist of variables and coefficients arranged in terms that can have exponents. Each term in a polynomial is made up of a coefficient (a number) and a variable raised to an exponent. Here's a breakdown:

• The expression in the problem \(9y^2 - 3y + 27\) is a polynomial.
• It has three terms: \(9y^2\), \(-3y\), and \(+27\).
• These terms vary in their degree, which is the highest exponent on the variable; for example, \(9y^2\) has a degree of 2.

Polynomials like this are frequent in algebra because they can represent a wide range of situations, from simple arithmetic to more complex models in calculus. Understanding how to work with them helps simplify the solving of mathematical problems. By using the distributive property to multiply polynomials, you ensure that each interaction between the terms is properly calculated and organized.

Algebraic Expressions

Algebraic expressions are any mathematical phrases that involve numbers, variables, and operations, such as addition or multiplication. They are used to represent situations in a general form that can be manipulated using algebraic methods.

• An algebraic expression becomes an equation if an equal sign is present.
• They can range from simple terms like \(3x\) to complex expressions like \(9y^2 - 3y + 27\).
• In the context of the exercise, \(-13(9y^2 - 3y + 27)\), it's an algebraic expression that we simplified using multiplication through the distributive property.

Mastering algebraic expressions involves knowing how to perform operations using properties such as the distributive property, combining like terms, and understanding variable manipulation. These skills are crucial for solving equations and simplifying mathematical expressions in algebra.