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Sagot :
To determine which pair of expressions can represent the length and width of Josephine's rectangular garden, we need to check which pairs multiply together to give us the area polynomial [tex]\(2x^2 + x - 6\)[/tex] square feet.
Let's break down and verify each pair individually:
### Pair 1: Length = [tex]\(x^2 - 3\)[/tex], Width = 2
Let's check the product of these two expressions:
[tex]\[ (x^2 - 3) \cdot 2 = 2x^2 - 6 \][/tex]
This product results in [tex]\(2x^2 - 6\)[/tex], which is not equal to the given area [tex]\(2x^2 + x - 6\)[/tex]. Therefore, this pair is not a valid representation of the length and width.
### Pair 2: Length = [tex]\(2x + 3\)[/tex], Width = [tex]\(x - 2\)[/tex]
Let's check the product of these two expressions:
[tex]\[ (2x + 3) \cdot (x - 2) \][/tex]
Using the distributive property (also known as FOIL for binomials), we expand this product:
[tex]\[ 2x \cdot x + 2x \cdot -2 + 3 \cdot x + 3 \cdot -2 = 2x^2 - 4x + 3x - 6 \][/tex]
Combining like terms, we get:
[tex]\[ 2x^2 - x - 6 \][/tex]
This product results in [tex]\(2x^2 - x - 6\)[/tex], which is not equal to the given area [tex]\(2x^2 + x - 6\)[/tex]. Therefore, this pair is not a valid representation either.
### Pair 3: Length = [tex]\(2x + 2\)[/tex], Width = [tex]\(x - 3\)[/tex]
Let's check the product of these two expressions:
[tex]\[ (2x + 2) \cdot (x - 3) \][/tex]
Using the distributive property, we expand this product:
[tex]\[ 2x \cdot x + 2x \cdot -3 + 2 \cdot x + 2 \cdot -3 = 2x^2 - 6x + 2x - 6 \][/tex]
Combining like terms, we get:
[tex]\[ 2x^2 - 4x - 6 \][/tex]
This product results in [tex]\(2x^2 - 4x - 6\)[/tex], which is also not equal to the given area [tex]\(2x^2 + x - 6\)[/tex]. Thus, this pair is not a valid representation.
### Pair 4: Length = [tex]\(2x - 3\)[/tex], Width = [tex]\(x + 2\)[/tex]
Let's check the product of these two expressions:
[tex]\[ (2x - 3) \cdot (x + 2) \][/tex]
Using the distributive property, we expand this product:
[tex]\[ 2x \cdot x + 2x \cdot 2 + -3 \cdot x + -3 \cdot 2 = 2x^2 + 4x - 3x - 6 \][/tex]
Combining like terms, we get:
[tex]\[ 2x^2 + x - 6 \][/tex]
This product results in [tex]\(2x^2 + x - 6\)[/tex], which is exactly equal to the given area polynomial. Therefore, this pair is a valid representation of the length and width.
### Conclusion
The only correct pair of expressions that represent the length and width of Josephine's rectangular garden, given an area of [tex]\(2x^2 + x - 6\)[/tex] square feet, is:
Length = [tex]\(2x - 3\)[/tex] feet, Width = [tex]\(x + 2\)[/tex] feet.
Let's break down and verify each pair individually:
### Pair 1: Length = [tex]\(x^2 - 3\)[/tex], Width = 2
Let's check the product of these two expressions:
[tex]\[ (x^2 - 3) \cdot 2 = 2x^2 - 6 \][/tex]
This product results in [tex]\(2x^2 - 6\)[/tex], which is not equal to the given area [tex]\(2x^2 + x - 6\)[/tex]. Therefore, this pair is not a valid representation of the length and width.
### Pair 2: Length = [tex]\(2x + 3\)[/tex], Width = [tex]\(x - 2\)[/tex]
Let's check the product of these two expressions:
[tex]\[ (2x + 3) \cdot (x - 2) \][/tex]
Using the distributive property (also known as FOIL for binomials), we expand this product:
[tex]\[ 2x \cdot x + 2x \cdot -2 + 3 \cdot x + 3 \cdot -2 = 2x^2 - 4x + 3x - 6 \][/tex]
Combining like terms, we get:
[tex]\[ 2x^2 - x - 6 \][/tex]
This product results in [tex]\(2x^2 - x - 6\)[/tex], which is not equal to the given area [tex]\(2x^2 + x - 6\)[/tex]. Therefore, this pair is not a valid representation either.
### Pair 3: Length = [tex]\(2x + 2\)[/tex], Width = [tex]\(x - 3\)[/tex]
Let's check the product of these two expressions:
[tex]\[ (2x + 2) \cdot (x - 3) \][/tex]
Using the distributive property, we expand this product:
[tex]\[ 2x \cdot x + 2x \cdot -3 + 2 \cdot x + 2 \cdot -3 = 2x^2 - 6x + 2x - 6 \][/tex]
Combining like terms, we get:
[tex]\[ 2x^2 - 4x - 6 \][/tex]
This product results in [tex]\(2x^2 - 4x - 6\)[/tex], which is also not equal to the given area [tex]\(2x^2 + x - 6\)[/tex]. Thus, this pair is not a valid representation.
### Pair 4: Length = [tex]\(2x - 3\)[/tex], Width = [tex]\(x + 2\)[/tex]
Let's check the product of these two expressions:
[tex]\[ (2x - 3) \cdot (x + 2) \][/tex]
Using the distributive property, we expand this product:
[tex]\[ 2x \cdot x + 2x \cdot 2 + -3 \cdot x + -3 \cdot 2 = 2x^2 + 4x - 3x - 6 \][/tex]
Combining like terms, we get:
[tex]\[ 2x^2 + x - 6 \][/tex]
This product results in [tex]\(2x^2 + x - 6\)[/tex], which is exactly equal to the given area polynomial. Therefore, this pair is a valid representation of the length and width.
### Conclusion
The only correct pair of expressions that represent the length and width of Josephine's rectangular garden, given an area of [tex]\(2x^2 + x - 6\)[/tex] square feet, is:
Length = [tex]\(2x - 3\)[/tex] feet, Width = [tex]\(x + 2\)[/tex] feet.
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