To represent the quadratic polynomial [tex]\(2x^2 + x - 6\)[/tex] using algebra tiles and determine its equivalent factored form, let's break it down into detailed steps. Also, we will determine the number of zero pairs needed to model this polynomial.
### Step-by-Step Solution:
1. Understand the Polynomial:
- The given polynomial is [tex]\(2x^2 + x - 6\)[/tex].
- Notice it’s a quadratic polynomial.
2. Identify Algebra Tiles:
- Algebra tiles are visual tools used to represent polynomials.
- We typically use:
- [tex]\(x^2\)[/tex] tiles for [tex]\(x^2\)[/tex] terms.
- [tex]\(x\)[/tex] tiles for [tex]\(x\)[/tex] terms.
- Unit tiles for constant terms.
3. Represent [tex]\(2x^2 + x - 6\)[/tex] using Algebra Tiles:
- [tex]\(2x^2\)[/tex]: This is represented by 2 [tex]\(x^2\)[/tex] tiles.
- [tex]\(x\)[/tex]: This is represented by 1 [tex]\(x\)[/tex] tile.
- [tex]\(-6\)[/tex]: This is represented by 6 unit tiles. Since it’s negative, we consider negative unit tiles here.
4. Determine the Factored Form:
- The polynomial can be factored, meaning we find two binomials whose product gives us the original polynomial.
5. Factored Form:
- The factored form of [tex]\(2x^2 + x - 6\)[/tex] is [tex]\((x + 2)(2x - 3)\)[/tex].
6. Number of Zero Pairs:
- Zero pairs in algebra tiles help to maintain the balance when simplifying or factoring.
- Zero pairs are when a positive tile and a negative tile cancel each other out. They are often used to transition between different forms or simplify the process in algebra tiles.
In this specific problem, though, the exact number of zero pairs may depend on the method used to transition from given polynomial to its factors using algebra tiles. So the exact number of zero pairs is undetermined without actually laying out the tiles and simplifying. However, this is not explicitly required to find the factored forms.
### Final Answer:
1. The equivalent factored form of [tex]\(2x^2 + x - 6\)[/tex] is [tex]\((x + 2)(2x - 3)\)[/tex].
2. The number of zero pairs needed to model this polynomial is context-dependent and not explicitly specified here.
So the blank spaces can be filled as follows:
- The equivalent factored form is [tex]\((x + 2)(2x - 3)\)[/tex].
- The number of zero pairs required is generally determined by the method of simplification. If unspecified, this can simply be labeled as context-dependent or undetermined based on algebra tile manipulation.
In summary, the polynomial [tex]\(2x^2 + x - 6\)[/tex] factors to [tex]\((x + 2)(2x - 3)\)[/tex].
