Problem 31 - Beginning and Intermediate Algebra

Question

Perform the indicated operations and simplify. $$\left(5 y^{3}-y^{2}+8 y+1\right)(3 y-4)$$

Step-by-Step Solution

Answer

The simplified result of the indicated operation is \(15y^4 - 23y^3 + 28y^2 - 29y - 4\).

Step 1: Distribute terms of second polynomial

To start, we will distribute each term from the second polynomial to each term of the first polynomial. We have: $$ (3y)(5y^3) + (-4)(5y^3) + (3y)(-y^2) + (-4)(-y^2) + (3y)(8y) + (-4)(8y) + (3y)(1) + (-4)(1) $$

Step 2: Multiply terms

Now, we will multiply the terms. $$ 15y^4 - 20y^3 - 3y^3 + 4y^2 + 24y^2 - 32y + 3y - 4 $$

Step 3: Combine like terms

Lastly, we simplify by combining like terms. $$ 15y^4 + (-20y^3 - 3y^3) + (4y^2 + 24y^2) + (-32y + 3y) - 4 $$ $$ 15y^4 - 23y^3 + 28y^2 - 29y - 4 $$ So, the simplified result is: $$ \boxed{15y^4 - 23y^3 + 28y^2 - 29y - 4} $$

Key Concepts

Distributive Property

The Distributive Property is a fundamental rule in algebra that allows you to multiply a single term by each term within a parenthesis. This property is essential in polynomial multiplication. For instance, when you have an expression like

• \(a(b + c)\), it can be rewritten as \(ab + ac\)

This property helps you break down complex expressions into simpler parts, making calculations easier.

In our exercise, we apply the Distributive Property by taking each term of the second polynomial \((3y - 4)\) and multiplying it by every term in the first polynomial \((5y^3 - y^2 + 8y + 1)\). This step is the foundation that allows you to later combine and simplify terms effectively.

Combining Like Terms

Combining like terms is the next important step after using the Distributive Property. In polynomial expressions, 'like terms' are those that have the same variable raised to the same power.

For example:

• \(3y^2\) and \(5y^2\) are like terms, but \(3y^2\) and \(5y\) are not.

When you have an expression like

• \(4y^3 + 2y^3\), you combine it to get \(6y^3\).

This process helps streamline expressions, making them easier to handle and interpret. In our exercise, after multiplying each pair of terms, we combine the results by polynomial terms:

• \(-20y^3\) and \(-3y^3\) become \(-23y^3\)
• \(4y^2\) and \(24y^2\) become \(28y^2\)

This step is crucial for obtaining the simplified polynomial.

Polynomial Simplification

Polynomial simplification involves two main processes: distributing terms and then combining like terms. These steps lead to a polynomial that is in its simplest form, meaning it has no like terms left to combine.

After distributing each term and combining like terms, you are left with a clean and concise expression.

• For example, the expression \(15y^4 - 20y^3 - 3y^3 + 4y^2 + 24y^2 - 32y + 3y - 4\) becomes \(15y^4 - 23y^3 + 28y^2 - 29y - 4\).

The absence of further like terms means the polynomial is fully simplified. Simplification is essential because it provides the most user-friendly and manageable form of the expression, useful for further calculations or evaluations.

Intermediate Algebra

Intermediate Algebra involves the application of algebraic concepts including polynomial operations, which is an integral part of mastering this level of math. By understanding operations such as distributing, multiplying, and combining terms, one can solve various algebraic expressions and equations with proficiency. This knowledge is critical for advancing in mathematics.

In the exercise provided, mastering these skills lets you simplify and manipulate complex expressions efficiently. Understanding Intermediate Algebra concepts like these services as a building block for higher-level mathematics, ensuring students have a strong foundation in algebraic thinking.