Question

write the expression then simplify.
1
if $3x - 2$ pieces of candy need to be distributed to $x + 6$ children, how many pieces of candy will be needed? write your answer as a polynomial in standard form.
2
find the product of three consecutive even numbers.

Explanation:

Problem 1

Step1: Understand the problem

We need to find the total number of candies when each of \(x + 6\) children gets \(3x - 2\) candies. So we multiply the number of candies per child by the number of children.
Expression: \((3x - 2)(x + 6)\)

Step2: Expand the product

Using the distributive property (FOIL method):
\[

$$\begin{align*} (3x - 2)(x + 6)&=3x\times x+3x\times6-2\times x - 2\times6\\ &=3x^{2}+18x-2x - 12 \end{align*}$$

\]

Step3: Combine like terms

Combine the \(x\) terms:
\(3x^{2}+(18x - 2x)-12=3x^{2}+16x - 12\)

Step1: Represent the consecutive even numbers

Let the first even number be \(2n\) (where \(n\) is an integer). Then the next consecutive even number is \(2n + 2\) and the third is \(2n+4\).

Step2: Find the product

The product of the three consecutive even numbers is \((2n)(2n + 2)(2n + 4)\)

First, factor out a 2 from each term:
\(2n\times2(n + 1)\times2(n + 2)=8n(n + 1)(n + 2)\)

We can also expand this:
First, multiply \(n(n + 1)(n + 2)=n(n^{2}+3n + 2)=n^{3}+3n^{2}+2n\)
Then multiply by 8: \(8(n^{3}+3n^{2}+2n)=8n^{3}+24n^{2}+16n\)

Answer:

\(3x^{2}+16x - 12\)

Problem 2