91Ó°ÊÓ

When adding polynomials, it’s important to identify and combine like terms. Like terms are terms that contain the same variable raised to the same power. They can be added or subtracted because they represent the same type of quantity.
In our exercise, \( -5y^3 \) and \( 8y^3 \) are like terms because they both contain \( y^3 \). Similarly, \( 3y \) and \( -4y \) are like terms because they both contain \( y \).
To make the process easier, list like terms in columns, aligning terms with matching variables and exponents. This helps you see easily which terms to combine and ensures you don't miss any terms.
Here's what alignment looks like:
\[ \begin{array}{r} -5y^3 + 3y \ \ 8y^3 - 4y \ \ \hline \end{array} \]
It becomes immediately clear which terms are like and should be combined together. This step is crucial for correct polynomial addition.

Coefficients are the numerical parts of the terms in polynomials. They are the numbers that multiply the variable(s). When adding like terms, you are essentially adding their coefficients.
In our exercise, the coefficient for \( y^3 \) in the first polynomial is -5. For the second polynomial, it's 8. By adding these coefficients together, \(-5 + 8 = 3\), we get \( 3y^3 \).
For the \( y \) terms, the coefficients are 3 and -4. Adding them gives \(-3y + -4y = -y\).
Remember:

• Only add or subtract coefficients of like terms.
• The variable and its exponent should remain unchanged after combining their coefficients.

Therefore, \( y^3 \) terms combine to \( 3y^3 \), and \( y \) terms combine to \( -y. \)

Simplifying expressions involves combining like terms to create the simplest form of the polynomial. After adding the coefficients of like terms, write the simplified polynomial without extra terms or errors.
In our example, we combined like terms and got \( 3y^3 - y \). That’s the simplified form of the expression.
Always check your final expression to ensure:

• All like terms are combined.
• There are no redundant terms.

Breaking the task down into smaller steps helps ensure nothing is missed and makes the process easier to follow.
The result should always be clean and easy to read, with all like terms combined properly into a single term for each variable and power involved.