Problem 42 - Algebra 1: Concepts and Skills

Question

Use a horizontal format to add or subtract. $$ 12-\left(y^{3}+10 y+16\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-y^{3} - 10y - 4\)

Step 1: Identify the bracketed term

In the given algebraic expression \(12-\left(y^{3}+10 y+16\right)\), the bracketed term is \(\left(y^{3}+10 y+16\right)\).

Step 2: Apply the minus operator to each term in the bracket

Applying the minus operator to each term in the bracket, we obtain \(12 - y^{3} - 10 y - 16\). This is because subtracting a number is equivalent to adding its opposite.

Step 3: Simplify the resulting expression

Finally, let's simplify. The numbers without any variables, 12 and -16, can be combined. This gives us \(-y^{3} - 10y + (-4)\).

Step 4: Final Result

The final simplified expression is \(-y^{3} - 10y - 4\)

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions involves reducing an expression to its simplest form. This means eliminating any unnecessary parentheses and combining like terms. To simplify expressions like the one in the problem, focus on these steps:

• Identify Like Terms: Like terms are terms that have the same variable part. For example, in the expression \(x^2 + 3x^2 + 2\), the like terms are \(x^2\) and \(3x^2\). They can be combined by adding their coefficients.
• Apply Operations Carefully: Whenever you apply a subtraction or addition operation, attend to each term separately. For instance, switching from subtraction to addition of negative terms should be handled with caution.
• Reorder Terms: It's sometimes helpful to rearrange terms, grouping like terms together to make simplification easier.

When we applied these principles to the exercise, we turned \(12 - y^3 - 10y - 16\) into the simplified form \(-y^3 - 10y - 4\). This shows that understanding the structure of your expression allows you to find its simplest form effectively.

Subtraction of Polynomials

Subtracting polynomials requires understanding how to deal with the minus operator. This operator flips the sign of every term in the polynomial that follows it. In the provided solution, we explored how subtracting the polynomial \(y^3 + 10y + 16\) from 12 works:

• Distribute the Minus Sign: When you see a negative sign before a polynomial wrapped in parentheses, apply the minus sign to each term inside. For example, \(- (y^3 + 10y + 16)\) becomes \(-y^3 - 10y - 16\).
• Combine Polynomials Carefully: Once the sign has been distributed, combine like terms or constants together. This is how 12 and -16 were simplified to \(-4\).
• Rewriting Terms: Ensure all terms are ordered, typically starting with the highest power of the variable first. This helps in easily identifying like terms.

By managing the negative signs properly, you can avoid common mistakes, ensuring your expressions are simplified correctly and reflect the right subtraction results.

Negative Numbers

Negative numbers often bring an added layer of complexity to algebraic expressions, particularly when subtraction is involved. Understanding how negative numbers work is critical for success in algebra.

• Concept of Opposites: Remember that subtracting a number is equivalent to adding its opposite. This means \(12 - 16\) is the same as \(12 + (-16)\), resulting in -4.
• Negative Coefficients: In our expression, applying a subtraction across the terms inside the parentheses resulted in negative coefficients. \(y^3\) became \(-y^3\), showing how subtraction affects both isolated numbers and those attached to variables.
• Operations with Negatives: Be mindful of how negatives interact with each other. Two negative numbers multiplied become positive, while adding two negatives simply results in a larger negative.

In our exercise, mastering these features of negative numbers allowed for accurate simplification, ensuring calculations were both clear and correct.