91Ó°ÊÓ

Polynomial subtraction might sound tricky, but it's a simple process when you break it down. Just like regular subtraction, you start by lining up the expressions. The main thing to remember is to subtract each term in the second polynomial from the corresponding term in the first polynomial. To solve the problem, you start with the expression \(10y^2 - 8y - 20\) and subtract \(5y^2 - y + 12\).

Ensure each term in both polynomials is in the same order (usually by descending powers, from highest to lowest).
This makes the subtraction process straightforward, as you're only dealing with like terms (more on that in the next section).
So, to recap, the step-by-step approach involves:

• Arranging the polynomials in order.
• Subtracting each term in the second polynomial from the corresponding term in the first polynomial.

Easy, right?

After arranging the polynomials properly, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, in our case, \(10y^2\) and \(5y^2\) are like terms, because both have the variable \(y\) raised to the power of 2.

Here’s how you combine like terms in a polynomial subtraction:

• Identify the like terms in both polynomials. For example, \(10y^2 - 8y - 20\) and \(5y^2 - y + 12\) both contain quadratic terms (terms with \(y^2\)), linear terms (terms with \(y\)), and constant terms (numbers without variables).
• You then subtract these like terms from each other:
• \begin{itemize}
• \(10y^2 - 5y^2 = 5y^2\)
• \(-8y - (-y) = -8y + y = -7y\)
• \(-20 - 12 = -32\)

The result of combining these like terms is \(5y^2 - 7y - 32\).

The distributive property is crucial when it comes to polynomial subtraction. This property states that you need to multiply everything inside the parentheses by the negative sign if you are subtracting polynomials.
For our example, \[ (10y^2 - 8y - 20) - (5y^2 - y + 12) \], you need to distribute the negative sign to each term in the second polynomial:

• Negative of \(5y^2\) is \(-5y^2\).
• Negative of \( - y\) is \(+(+ y)\) materializing finally to a positive term \(y\).
• Negative of \(12\) is \(-12\).

This distribution changes the expression to:
\begin{equation*}(10y^2 - 8y - 20) - 5y^2 + y - 12ewline (10y^2 - 5y^2) + (-8y + y) + (-20 - 12) = 5y^2 - 7y - 32
And that's how the distributive property simplifies your polynomial subtraction!