Question

Add, subtract, or multiply, as indicated. Express your answer as a single polynomial in standard form. $$ 8\left(1-y^{3}\right)+4\left(1+y+y^{2}+y^{3}\right) $$

Short answer

12 + 4y + 4y^{2} - 4y^{3}

Step-by-step solution

- Distribute the 8

Apply the distributive property to the first term: \[ 8(1-y^{3}) = 8 \times 1 + 8 \times (-y^{3}) = 8 - 8y^{3} \]

- Distribute the 4

Apply the distributive property to the second term: \[ 4(1+y+y^{2}+y^{3}) = 4 \times 1 + 4 \times y + 4 \times y^{2} + 4 \times y^{3} = 4 + 4y + 4y^{2} + 4y^{3} \]

- Combine like terms

Combine the results from steps 1 and 2: \[ (8 - 8y^{3}) + (4 + 4y + 4y^{2} + 4y^{3}) \] Simplify the expression by combining like terms: \[ 8 + 4 - 8y^{3} + 4y^{3} + 4y + 4y^{2} \] Group the terms: \[ (8+4) + (4y) + (4y^{2}) + (-8y^{3} + 4y^{3}) \] Resulting in: \[ 12 + 4y + 4y^{2} - 4y^{3} \]

Key concepts

distributive property

The distributive property is a fundamental principle in algebra that lets you multiply a sum by distributing the multiplication to each term inside the parentheses. For example, in the expression \(8(1 - y^3)\), we distribute the 8 to both 1 and \(-y^3\).
This results in: \ \(8 \times 1 + 8 \times (-y^3) = 8 - 8y^3\).
Similarly, in the expression \(4(1 + y + y^2 + y^3)\), we distribute the 4 to each term: \ \(4 \times 1 + 4 \times y + 4 \times y^2 + 4 \times y^3 = 4 + 4y + 4y^2 + 4y^3\).
Understanding the distributive property makes it easier to simplify expressions and solve equations.

combining like terms

After distributing, we often end up with terms that are similar. These terms can be combined to simplify the expression further. In this problem, we first applied the distributive property and obtained the terms \(8 - 8y^3\) and \(4 + 4y + 4y^2 + 4y^3\).
To combine like terms, we add or subtract coefficients of the same variables.
Here’s how we do it:
1. List out all the terms: \(8, -8y^3, 4, 4y, 4y^2, 4y^3\).
2. Group the terms with the same variable: \ \((8 + 4) + (4y) + (4y^2) + (-8y^3 + 4y^3)\).
3. Combine them: \ \(12 + 4y + 4y^2 - 4y^3\).

By combining like terms, the expression becomes simpler and easier to read.

standard form of a polynomial

The standard form of a polynomial arranges the terms of the polynomial in descending order of their exponent values.
In our example, after combining like terms, the polynomial is \(12 + 4y + 4y^2 - 4y^3\).
The polynomial should be written in the order of the highest power of the variable to the lowest:
\ \(-4y^3 + 4y^2 + 4y + 12\).
Writing polynomials in standard form makes it easier to compare polynomials, solve polynomial equations, and communicate mathematical ideas more clearly.

Always make sure that your final step is to write the polynomial in this format, ensuring clarity and consistency in your math solutions.