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Sagot :
Let's determine the product of [tex]\((x - 2)(x - 3)\)[/tex] step by step and see if Cherise's final result is [tex]\(x^2 - 5x - 6\)[/tex].
We can solve this by expanding the expression:
1. Distribute each term in the first binomial to each term in the second binomial:
[tex]\[ (x - 2)(x - 3) = x(x - 3) + (-2)(x - 3) \][/tex]
2. Apply the distributive property (also known as the FOIL method) to each term:
- First, multiply [tex]\(x\)[/tex] by [tex]\(x\)[/tex] and then [tex]\(x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[ x \cdot x + x \cdot (-3) = x^2 - 3x \][/tex]
- Next, multiply [tex]\(-2\)[/tex] by [tex]\(x\)[/tex] and then [tex]\(-2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[ -2 \cdot x + (-2) \cdot (-3) = -2x + 6 \][/tex]
3. Combine all the terms together:
[tex]\[ x^2 - 3x - 2x + 6 \][/tex]
4. Simplify the expression by combining like terms (combine the [tex]\(x\)[/tex]-terms):
[tex]\[ x^2 - 3x - 2x + 6 = x^2 - 5x + 6 \][/tex]
Therefore, the correct product of [tex]\((x - 2)(x - 3)\)[/tex] is indeed [tex]\(x^2 - 5x + 6\)[/tex].
Now let's analyze Cherise's representation based on the possible mistakes:
1. Incorrect multiplication of the [tex]\(x\)[/tex]-tiles by the negative integer tiles: This means Cherise made an error in multiplying [tex]\(x\)[/tex] by [tex]\(-3\)[/tex] and/or [tex]\(-2\)[/tex] by [tex]\(x\)[/tex]. But our distribution indicates [tex]\(x \cdot (-3) = -3x\)[/tex] and [tex]\(-2 \cdot x = -2x\)[/tex] which are correct.
2. Incorrect multiplication of the negative integer tiles by the other negative integer tiles: This refers to a mistake in multiplying [tex]\(-2\)[/tex] by [tex]\(-3\)[/tex]. Our result shows [tex]\(-2 \cdot (-3) = 6\)[/tex], which is correct.
3. Incorrect addition of the terms together: Cherise might have made a mistake in combining the terms. Combining [tex]\(-3x\)[/tex] and [tex]\(-2x\)[/tex] correctly gives [tex]\(-5x\)[/tex]. However, if she incorrectly added these terms, she could have gotten the wrong middle term.
4. Correct representation: Her final result is indeed [tex]\(x^2 - 5x + 6\)[/tex], which matches our derived product.
Based on the given result and our calculations, we conclude:
Yes, the product is [tex]\(x^2 - 5x + 6\)[/tex].
We can solve this by expanding the expression:
1. Distribute each term in the first binomial to each term in the second binomial:
[tex]\[ (x - 2)(x - 3) = x(x - 3) + (-2)(x - 3) \][/tex]
2. Apply the distributive property (also known as the FOIL method) to each term:
- First, multiply [tex]\(x\)[/tex] by [tex]\(x\)[/tex] and then [tex]\(x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[ x \cdot x + x \cdot (-3) = x^2 - 3x \][/tex]
- Next, multiply [tex]\(-2\)[/tex] by [tex]\(x\)[/tex] and then [tex]\(-2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[ -2 \cdot x + (-2) \cdot (-3) = -2x + 6 \][/tex]
3. Combine all the terms together:
[tex]\[ x^2 - 3x - 2x + 6 \][/tex]
4. Simplify the expression by combining like terms (combine the [tex]\(x\)[/tex]-terms):
[tex]\[ x^2 - 3x - 2x + 6 = x^2 - 5x + 6 \][/tex]
Therefore, the correct product of [tex]\((x - 2)(x - 3)\)[/tex] is indeed [tex]\(x^2 - 5x + 6\)[/tex].
Now let's analyze Cherise's representation based on the possible mistakes:
1. Incorrect multiplication of the [tex]\(x\)[/tex]-tiles by the negative integer tiles: This means Cherise made an error in multiplying [tex]\(x\)[/tex] by [tex]\(-3\)[/tex] and/or [tex]\(-2\)[/tex] by [tex]\(x\)[/tex]. But our distribution indicates [tex]\(x \cdot (-3) = -3x\)[/tex] and [tex]\(-2 \cdot x = -2x\)[/tex] which are correct.
2. Incorrect multiplication of the negative integer tiles by the other negative integer tiles: This refers to a mistake in multiplying [tex]\(-2\)[/tex] by [tex]\(-3\)[/tex]. Our result shows [tex]\(-2 \cdot (-3) = 6\)[/tex], which is correct.
3. Incorrect addition of the terms together: Cherise might have made a mistake in combining the terms. Combining [tex]\(-3x\)[/tex] and [tex]\(-2x\)[/tex] correctly gives [tex]\(-5x\)[/tex]. However, if she incorrectly added these terms, she could have gotten the wrong middle term.
4. Correct representation: Her final result is indeed [tex]\(x^2 - 5x + 6\)[/tex], which matches our derived product.
Based on the given result and our calculations, we conclude:
Yes, the product is [tex]\(x^2 - 5x + 6\)[/tex].
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