91Ó°ÊÓ

When multiplying polynomials, the distributive property is an essential tool. It allows you to multiply a single term by each term within a parenthesis. For example, in the given problem, we use the distributive property to multiply each term in the binomial \( y - 6 \) with each term in the trinomial \( y^2 - 10y + 9 \). This step-by-step process is broken down as follows:
First, multiply \( y \) by each term in the trinomial:
\[ y \times (y^2 - 10y + 9) = y \times y^2 + y \times (-10y) + y \times 9 = y^3 - 10y^2 + 9y \]
Next, multiply \( -6 \) by each term in the trinomial:
\[-6 \] \(([y^2 - 10y + 9])\) \[= -6 \times y^2 + (-6) \times (-10y) + (-6) \times 9 = -6y^2 + 60y - 54 \]
This systematic approach ensures that every term in the binomial is correctly distributed across each term in the trinomial.

After distributing, you will have several terms that need to be combined. Combining like terms means adding or subtracting coefficients of terms that have the same variable raised to the same power. From the steps in the exercise, we end up with the following terms:
\[ y^3 - 10y^2 + 9y - 6y^2 + 60y - 54 \]
Now identify like terms:

• \( y^3 \)
• \( -10y^2 \) and \( -6y^2 \)
• \( 9y \) and \( 60y \)

Combine them:
\[-10y^2 - 6y^2 = -16y^2, \]
\[9y + 60y = 69y \]
Thus, the final expression becomes:
\[y^3 - 16y^2 + 69y - 54 \]
This step is crucial as it simplifies the polynomial into its final form.

Understanding binomials and trinomials is key in polynomial multiplication. A binomial is a polynomial with two terms, like \(y - 6\), whereas a trinomial has three terms, such as \(y^2 - 10y + 9\). When multiplying a binomial by a trinomial, you use the distributive property to ensure each term in the binomial is multiplied by each term in the trinomial.
For instance:

• Multiply first term of binomial (\(y\)) with each term of the trinomial:
• Then multiply second term of binomial (\(-6\)) with each term of the trinomial:

This results in six individual multiplications, which are then combined and simplified by combining like terms. This is what leads us to our final simplified polynomial result.