Answer :
To determine the factors of the polynomial from the given pairs, we need to multiply each pair and see which pair results in the original polynomial. Here are the steps for finding the factors:
### Pair 1: [tex]\((x-1)\)[/tex] and [tex]\((x+3)\)[/tex]
Let's multiply these two factors:
[tex]\[ (x-1)(x+3) \][/tex]
Apply the distributive property (also known as FOIL):
[tex]\[ (x-1)(x+3) = x(x+3) - 1(x+3) \][/tex]
[tex]\[ = x^2 + 3x - x - 3 \][/tex]
[tex]\[ = x^2 + 2x - 3 \][/tex]
### Pair 2: [tex]\((x+1)\)[/tex] and [tex]\((x-3)\)[/tex]
Let's multiply these two factors:
[tex]\[ (x+1)(x-3) \][/tex]
Apply the distributive property:
[tex]\[ (x+1)(x-3) = x(x-3) + 1(x-3) \][/tex]
[tex]\[ = x^2 - 3x + x - 3 \][/tex]
[tex]\[ = x^2 - 2x - 3 \][/tex]
### Pair 3: [tex]\((x-2)\)[/tex] and [tex]\((x+3)\)[/tex]
Let's multiply these two factors:
[tex]\[ (x-2)(x+3) \][/tex]
Apply the distributive property:
[tex]\[ (x-2)(x+3) = x(x+3) - 2(x+3) \][/tex]
[tex]\[ = x^2 + 3x - 2x - 6 \][/tex]
[tex]\[ = x^2 + x - 6 \][/tex]
### Pair 4: [tex]\((x+2)\)[/tex] and [tex]\((x-3)\)[/tex]
Let's multiply these two factors:
[tex]\[ (x+2)(x-3) \][/tex]
Apply the distributive property:
[tex]\[ (x+2)(x-3) = x(x-3) + 2(x-3) \][/tex]
[tex]\[ = x^2 - 3x + 2x - 6 \][/tex]
[tex]\[ = x^2 - x - 6 \][/tex]
Now that we have expanded all the pairs, the polynomial results are:
1. [tex]\((x-1)(x+3) = x^2 + 2x - 3\)[/tex]
2. [tex]\((x+1)(x-3) = x^2 - 2x - 3\)[/tex]
3. [tex]\((x-2)(x+3) = x^2 + x - 6\)[/tex]
4. [tex]\((x+2)(x-3) = x^2 - x - 6\)[/tex]
Based on these calculations:
- If the original polynomial is [tex]\( x^2 + 2x - 3 \)[/tex], the factors are [tex]\((x-1)\)[/tex] and [tex]\((x+3)\)[/tex].
- If the original polynomial is [tex]\( x^2 - 2x - 3\)[/tex], the factors are [tex]\((x+1)\)[/tex] and [tex]\((x-3)\)[/tex].
- If the original polynomial is [tex]\( x^2 + x - 6\)[/tex], the factors are [tex]\((x-2)\)[/tex] and [tex]\((x+3)\)[/tex].
- If the original polynomial is [tex]\( x^2 - x - 6\)[/tex], the factors are [tex]\((x+2)\)[/tex] and [tex]\((x-3)\)[/tex].
By knowing the original polynomial, we can determine its factors from the list above.
### Pair 1: [tex]\((x-1)\)[/tex] and [tex]\((x+3)\)[/tex]
Let's multiply these two factors:
[tex]\[ (x-1)(x+3) \][/tex]
Apply the distributive property (also known as FOIL):
[tex]\[ (x-1)(x+3) = x(x+3) - 1(x+3) \][/tex]
[tex]\[ = x^2 + 3x - x - 3 \][/tex]
[tex]\[ = x^2 + 2x - 3 \][/tex]
### Pair 2: [tex]\((x+1)\)[/tex] and [tex]\((x-3)\)[/tex]
Let's multiply these two factors:
[tex]\[ (x+1)(x-3) \][/tex]
Apply the distributive property:
[tex]\[ (x+1)(x-3) = x(x-3) + 1(x-3) \][/tex]
[tex]\[ = x^2 - 3x + x - 3 \][/tex]
[tex]\[ = x^2 - 2x - 3 \][/tex]
### Pair 3: [tex]\((x-2)\)[/tex] and [tex]\((x+3)\)[/tex]
Let's multiply these two factors:
[tex]\[ (x-2)(x+3) \][/tex]
Apply the distributive property:
[tex]\[ (x-2)(x+3) = x(x+3) - 2(x+3) \][/tex]
[tex]\[ = x^2 + 3x - 2x - 6 \][/tex]
[tex]\[ = x^2 + x - 6 \][/tex]
### Pair 4: [tex]\((x+2)\)[/tex] and [tex]\((x-3)\)[/tex]
Let's multiply these two factors:
[tex]\[ (x+2)(x-3) \][/tex]
Apply the distributive property:
[tex]\[ (x+2)(x-3) = x(x-3) + 2(x-3) \][/tex]
[tex]\[ = x^2 - 3x + 2x - 6 \][/tex]
[tex]\[ = x^2 - x - 6 \][/tex]
Now that we have expanded all the pairs, the polynomial results are:
1. [tex]\((x-1)(x+3) = x^2 + 2x - 3\)[/tex]
2. [tex]\((x+1)(x-3) = x^2 - 2x - 3\)[/tex]
3. [tex]\((x-2)(x+3) = x^2 + x - 6\)[/tex]
4. [tex]\((x+2)(x-3) = x^2 - x - 6\)[/tex]
Based on these calculations:
- If the original polynomial is [tex]\( x^2 + 2x - 3 \)[/tex], the factors are [tex]\((x-1)\)[/tex] and [tex]\((x+3)\)[/tex].
- If the original polynomial is [tex]\( x^2 - 2x - 3\)[/tex], the factors are [tex]\((x+1)\)[/tex] and [tex]\((x-3)\)[/tex].
- If the original polynomial is [tex]\( x^2 + x - 6\)[/tex], the factors are [tex]\((x-2)\)[/tex] and [tex]\((x+3)\)[/tex].
- If the original polynomial is [tex]\( x^2 - x - 6\)[/tex], the factors are [tex]\((x+2)\)[/tex] and [tex]\((x-3)\)[/tex].
By knowing the original polynomial, we can determine its factors from the list above.
