Problem 98 - Intermediate Algebra

Question

Simplify each expression. \(3\left(4 y^{3}+3 y-2\right)+2\left(3 y^{2}-y+3\right)-5\left(2 y^{3}-y^{2}-2\right)\)

Step-by-Step Solution

Answer

The simplified expression is \( 2y^3 + 11y^2 + 7y + 10 \).

Step 1: Expand Each Term Separately

We start by expanding each term in the expression individually. First, expand the expression within each parenthesis by distributing the number that multiplies each term.1. Expand: - For the first group: \[ 3(4y^3 + 3y - 2) = 3 \times 4y^3 + 3 \times 3y - 3 \times 2 = 12y^3 + 9y - 6 \] - For the second group: \[ 2(3y^2 - y + 3) = 2 \times 3y^2 - 2 \times y + 2 \times 3 = 6y^2 - 2y + 6 \] - For the third group: \[ -5(2y^3 - y^2 - 2) = -5 \times 2y^3 + 5 \times y^2 + 5 \times 2 = -10y^3 + 5y^2 + 10 \]

Step 2: Combine Like Terms

Next, combine like terms from the expanded expressions. Arrange the terms by the power of \( y \):Collect the terms for each degree of \( y \):- **Cubic terms (\( y^3 \))**: \[ 12y^3 - 10y^3 = 2y^3 \]- **Quadratic terms (\( y^2 \))**: \[ 6y^2 + 5y^2 = 11y^2 \]- **Linear terms (\( y \))**: \[ 9y - 2y = 7y \]- **Constant terms**: \[ -6 + 6 + 10 = 10 \]

Step 3: Write the Final Expression

Based on the combined terms, we write the simplified expression in order of decreasing powers of \( y \):\[ 2y^3 + 11y^2 + 7y + 10 \]

Key Concepts

Distributive Property

To simplify polynomial expressions, the distributive property is often the very first tool we use in our toolbox. This property allows you to multiply a single term across terms within a set of parentheses. Essentially, you are distributing the factor to each term separately.

For example, consider the expression \(3(4y^3 + 3y - 2)\). You distribute the 3 to each term inside the parentheses:

• Multiply 3 by \(4y^3\) to get \(12y^3\).
• Multiply 3 by \(3y\) to get \(9y\).
• Multiply 3 by \(-2\) to get \(-6\).

By applying the distributive property, you can see each part clearly expanded, which aids in the next steps of simplification.
If the expression involves subtraction or negative terms, it’s crucial to remember that you distribute the negative sign as well. This means changing the signs of each term in the parentheses accordingly.

Combining Like Terms

Once you've used the distributive property to expand an expression, the next step is to combine like terms. Like terms are terms in a polynomial that have the same variable raised to the same power. This allows you to simplify the expression by merging these terms together.

For instance, after expanding your expression, you might have terms like \(12y^3\) and \(-10y^3\). Since they both have \(y^3\), they are like terms and can be combined:

• Combine cubic terms: \(12y^3 - 10y^3 = 2y^3\).
• Combine quadratic terms: if you have terms like \(6y^2\) and \(5y^2\), you add: \(6y^2 + 5y^2 = 11y^2\).
• Combine linear terms: similarly, \(9y\) and \(-2y\) become \(7y\).
• Finally, don't forget the constants: \(-6 + 6 + 10 = 10\).

Combining like terms is akin to simplifying your grocery list by counting all the apples together and all the bananas together, making the list shorter and clearer.

Polynomial Expressions

Polynomial expressions are mathematical sentences where terms are coalesced by addition or subtraction. Each term in a polynomial can include variables with different exponents or coefficients, constants, or a combination of all.

Understanding the structure of polynomials is key to manipulation and simplification:

• Terms: Each component, like \(4y^3\) or \(-2\), is a term.
• Degrees: The power of the variable. In \(4y^3\), the degree is 3.
• Coefficients: The numerical part attached to a variable, like 4 in \(4y^3\).
• Constant: A term without a variable, for example, -2 or 10.

Polynomials show up everywhere in mathematics because they can express a wide range of functions and model various real-world situations. When simplifying polynomials, being aware of each term's properties helps in both understanding the expression and manipulating it effectively. After simplification, polynomial expressions are usually rewritten in canonical order, starting with the highest degree.