Problem 77 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ -5 y\left(y^{2}-3 y-6\right)-2 y\left(3 y^{2}+7\right)+(-2)(-5) $$

Step-by-Step Solution

Answer

Answer: $$-11y^3 + 15y^2 + 16y + 10$$

Step 1: Distribute the terms to their respective expressions inside the parentheses

We'll distribute -5y to (y^2 - 3y - 6) and distribute -2y to (3y^2 + 7): $$ -5y(y^2 - 3y - 6) - 2y(3y^2 + 7) + (-2)(-5) = -5y^3 + 15y^2 + 30y - 6y^3 - 14y + 10 $$

Step 2: Combine like terms

Now we'll group and combine the terms with the same degree: $$ = (-5y^3 - 6y^3) + (15y^2) + (30y - 14y) + 10 $$

Step 3: Simplify the expression

After combining like terms, we can simplify the expression: $$ = -11y^3 + 15y^2 + 16y + 10 $$ So, the simplified algebraic expression is $$ -11y^3 + 15y^2 + 16y + 10 $$

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical expression that contains variables, numbers, and operations. Expressions can include constants, coefficients, and variables. For example, in the expression \(-5y^3 + 15y^2 + 16y + 10\), there are:

• Variables: Represented here by \(y\), which can take on different values to satisfy the expression.
• Coefficients: These are numbers that multiply a variable, like \(-5\) in \(-5y^3\) or \(15\) in \(15y^2\).
• Constants: Numbers that stand alone without a variable, such as 10 in the example.

Algebraic expressions help us to model real-world problems by representing mathematical relationships succinctly. They are foundational in algebra because they form the basis for equations that we solve for unknown values.

Distributive Property

The distributive property is a crucial algebraic property that allows us to distribute multiplication over addition or subtraction inside parentheses. It is expressed algebraically as \(a(b + c) = ab + ac\). In the context of our problem, we applied the distributive property to the expressions:

• \(-5y(y^2 - 3y - 6)\) to expand and distribute \(-5y\) across each term inside the parentheses.
• \(-2y(3y^2 + 7)\) to distribute \(-2y\), multiplying each term separately.

This property is essential for simplifying expressions, especially when dealing with parentheses, as it allows us to break down complex expressions into simpler, more manageable terms.

Combining Like Terms

Combining like terms is a process used to simplify algebraic expressions by merging terms with the same variable and exponent. In our example, after distributing, we have a few terms that can be combined:

• \(-5y^3\) and \(-6y^3\) are combined because they both contain \(y^3\), resulting in \(-11y^3\).
• For the \(y^2\) terms, \(15y^2\) stays as is because there are no other \(y^2\) terms to combine with.
• \(30y\) and \(-14y\) are like terms that combine to give \(16y\).

This step is crucial because it leads to the most simplified form of an expression. Combining like terms reduces an expression to its simplest form, making further calculations more straightforward.

Polynomials

Polynomials are a specific type of algebraic expression that consist of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The example expression \(-11y^3 + 15y^2 + 16y + 10\) is a polynomial:

• The term \(-11y^3\) is a cubic term because the degree (highest power) of the variable \(y\) is 3.
• \(15y^2\) is a quadratic term as the power is 2.
• \(16y\) is a linear term with a power of 1.
• The number 10 is a constant term because it does not include any variables.

Polynomials are useful for modeling a variety of real-world processes and can be easily manipulated using arithmetic operations, making them fundamental in algebra and calculus.