To correctly factor the expression \( 3z^2 - 13z - 10 \), we can follow these ordered steps:
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Determine if there is a greatest common factor amongst all the terms.
- Check the coefficients for any common factors. In this case, the coefficients are 3, -13, and -10. There is no common factor among these coefficients.
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Find the two numbers that give a product of -30 and a sum of -13.
- We need to factor the expression \( 3z^2 - 13z - 10 \). The product of the coefficients of \( z^2 \) and the constant term is \( 3 \times -10 = -30 \). We need to find two numbers that multiply to -30 and add to -13. The numbers -15 and 2 work here (-15 * 2 = -30 and -15 + 2 = -13).
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Replace the middle term with the two numbers found -15 and 2.
- Rewrite the expression: \( 3z^2 - 15z + 2z - 10 \).
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Factor by grouping: Take the greatest common factor out of the first two terms and the greatest common factor out of the last two terms.
- Group the terms: \( (3z^2 - 15z) + (2z - 10) \).
- Factor out \( 3z \) from the first group and \( 2 \) from the second group:
- \( 3z(z - 5) + 2(z - 5) \).
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Factor out the common factor (z - 5) once more to get your final answer.
- Now, we can factor out \( (z - 5) \):
- \( (z - 5)(3z + 2) \).
- Now, we can factor out \( (z - 5) \):
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Final answer: \( (z - 5)(3z + 2) \).
So the ordered steps to factor \( 3z^2 - 13z - 10 \) are:
- Determine if there is a greatest common factor amongst all the terms.
- Find the two numbers that give a product of -30 and a sum of -13.
- Replace the middle term with -15 and 2.
- Factor by grouping.
- Factor out the common factor once more to get your final answer.
- Final answer: \( (z - 5)(3z + 2) \).
(Note: There were typographical errors in the original text, and some parts of the original procedure's text do not strictly correspond to the expression being factored, such as “Multiply 3x-10.” The correct expression is indeed \( 3z^2 - 13z - 10 \) not \( 3x - 10 \)).
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