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I. Multiplying Monomials and Binomials

When you multiply a monomial by a binomial, you are distributing the monomial to each term in the binomial. This means you multiply the monomial by each term in the binomial separately, then add the results together. For example, to multiply [tex]\(3x\)[/tex] by [tex]\(x + 4\)[/tex], you would do:

[tex]\[
\begin{aligned}
(3x)(x+4) &= (3x \cdot x) + (3x \cdot 4) \\
&= 3x^2 + 12x
\end{aligned}
\][/tex]

### Activity 1: Fill in the Blank

Fill in the blank with the correct words.

1. When multiplying a monomial by a binomial, you need to use the \_\_\_\_\_ property.
2. To multiply [tex]\(4x\)[/tex] by [tex]\(3x + 5\)[/tex], you first multiply [tex]\(4x\)[/tex] by \_\_\_\_\_.
3. The second step in multiplying [tex]\(4x\)[/tex] by [tex]\(3x + 5\)[/tex] is to multiply [tex]\(4x\)[/tex] by \_\_\_\_\_.
4. If you have [tex]\(2y\)[/tex] and you multiply it by [tex]\(y - 7\)[/tex], the result of [tex]\(2y \cdot y\)[/tex] is \_\_\_\_\_.
5. The final step in multiplying a monomial by a binomial is to \_\_\_\_\_ the results of each multiplication.



Answer :

GinnyAnswer
Let's walk through each of the questions one by one:

1. When multiplying a monomial by a binomial, you need to use the distributive property.
- The distributive property states that [tex]\(a(b + c) = ab + ac\)[/tex]. This property helps in distributing the monomial across each term in the binomial.

2. To multiply [tex]\(4x\)[/tex] by [tex]\(3x + 5\)[/tex], you first multiply [tex]\(4x\)[/tex] by 3x.
- When you begin multiplying the monomial [tex]\(4x\)[/tex] by the binomial [tex]\(3x + 5\)[/tex], you start by multiplying [tex]\(4x\)[/tex] by the first term in the binomial, which is [tex]\(3x\)[/tex].

3. The second step in multiplying [tex]\(4x\)[/tex] by [tex]\(3x + 5\)[/tex] is to multiply [tex]\(4x\)[/tex] by 5.
- After multiplying [tex]\(4x\)[/tex] by [tex]\(3x\)[/tex], the next step is to multiply [tex]\(4x\)[/tex] by the second term in the binomial, which is [tex]\(5\)[/tex].

4. If you have [tex]\(2y\)[/tex] and you multiply it by [tex]\(y - 7\)[/tex], the result of [tex]\((2y(y - 7))\)[/tex] is 2y^2 - 14y.
- Using the distributive property, you multiply [tex]\(2y\)[/tex] by [tex]\(y\)[/tex] to get [tex]\(2y^2\)[/tex], and then multiply [tex]\(2y\)[/tex] by [tex]\(-7\)[/tex] to get [tex]\(-14y\)[/tex]. Adding both results gives [tex]\(2y^2 - 14y\)[/tex].

5. The final step in multiplying a monomial by a binomial is to add the results of each multiplication.
- After performing the individual multiplications, you combine (add) the results to get the final expression.

So the completed answers should be:

1. When multiplying a monomial by a binomial, you need to use the distributive property.
2. To multiply [tex]\(4x\)[/tex] by [tex]\(3x + 5\)[/tex], you first multiply [tex]\(4x\)[/tex] by 3x.
3. The second step in multiplying [tex]\(4x\)[/tex] by [tex]\(3x + 5\)[/tex] is to multiply [tex]\(4x\)[/tex] by 5.
4. If you have [tex]\(2y\)[/tex] and you multiply it by [tex]\(y - 7\)[/tex], the result of [tex]\((2y(y - 7))\)[/tex] is 2y^2 - 14y.
5. The final step in multiplying a monomial by a binomial is to add the results of each multiplication.

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