91Ó°ÊÓ

Like terms are an essential concept in polynomial operations. They refer to terms within the polynomial that have the same variable raised to the same power. Essentially, their variable parts are identical. This similarity allows us to group them together for easier calculation.
In the given exercise, we have the polynomials: \((4y^2 + 10y + 3)\text{ and }(8y^2 - 6y + 5)\). Here, if we look closely, we identify the like terms:

• \(4y^2\ and\ 8y^2\)
• \(10y\ and\ -6y\)
• \(3\ and\ 5\)

The terms \(4y^2\) and \(8y^2\) are like terms because they both have \(y^2\). Similarly, \(10y\) and \(-6y\) are like terms because they both have \(y\). Finally, \(3\) and \(5\) are constants, so they are also considered like terms. Grouping these terms can make the process of combining them straightforward and manageable.

Once like terms have been identified, the next step is to combine their coefficients. The coefficient is the numerical factor in the term, and combining them involves adding or subtracting these numbers depending on their signs.
In our exercise, we combined the following coefficients:

• For \(y^2\) terms: \[4 + 8 = 12\]
• For \(y\) terms: \[10 - 6 = 4\]
• For constants: \[3 + 5 = 8\]

This step ensures that all like terms are merged into a single term, simplifying the polynomial. Adding the coefficients of like terms gives us the new coefficients for the combined terms, making the polynomial easier to manage and understand.

After combining the coefficients of like terms, we arrive at the resulting polynomial. In this exercise, we get the simplified polynomial by assembling all the new terms:
\[12y^2 + 4y + 8\]
Each part of this polynomial comes from the combined like terms:

• \(12y^2\) from \(4y^2 + 8y^2\)
• \(4y\) from \(10 - 6y\)
• \(8\) from \(3 + 5\)

This step finalizes the process and presents the simplified expression where all like terms have been successfully combined. Therefore, the resulting polynomial is a cleaner and more straightforward expression of the given polynomials' addition.