Combining like terms in a two-binomial polynomial sum

Key takeaway: This problem is a direct test of combining like terms. The key is to group $y^2$ terms, $y$ terms, and constants separately before writing the simplified polynomial.

Group like terms first

The expression is

$$(8y^2-4y)+(7y^2-9y+7).$$

A clean way to simplify is to remove the parentheses and then collect terms with the same variable power. Like terms must have the same variable part and exponent. That means $8y^2$ and $7y^2$ can combine, $-4y$ and $-9y$ can combine, and the constant $7$ stays by itself.

Combine each category

Write the expression in horizontal form:

$$8y^2-4y+7y^2-9y+7.$$

Now combine the quadratic terms:

$$8y^2+7y^2=15y^2.$$

Combine the linear terms:

$$-4y-9y=-13y.$$

The constant term is just $7$.

So the simplified result is

$$15y^2-13y+7.$$

Why the horizontal format matters

The phrase "use a horizontal format" means you should rewrite everything in one line before combining. This helps prevent sign errors and makes it easier to see which terms belong together. It is especially useful when one set of parentheses begins with a plus sign, because that means every sign inside the parentheses stays the same after removing them.

Common checks

A fast self-check is to verify that there are no remaining parentheses and no like terms left uncombined. The final answer should have one $y^2$ term, one $y$ term, and one constant. If you still see repeated powers of $y$, the expression is not fully simplified.

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Pitfalls the pros know 👇 A frequent mistake is changing signs when removing parentheses even though the expression has a plus sign between the groups. Since the problem is $(8y^2-4y)+(7y^2-9y+7)$, nothing inside the second parentheses changes sign. Another common error is combining $8y^2$ with $-4y$, but those are not like terms because the exponents differ. Keep powers separate: quadratic with quadratic, linear with linear, constant with constant. That habit prevents almost every simplification error on this kind of problem.

What if the problem changes? If the expression were $(8y^2-4y)-(7y^2-9y+7)$, then the subtraction sign would change every sign inside the second parentheses. The new expression would become $8y^2-4y-7y^2+9y-7$, and the simplified result would be $y^2+5y-7$. This variant shows why the operation between parentheses matters as much as the terms themselves.

Tags: like terms, distributive property, polynomial simplification