Question
Order the steps from first to last that would be used to factor: 3x² - 13x 10 (7 points) - = Determine if there is a greatest common factor amongst all the terms. = Factor the common factor once more to get your final answer. = Final answer: (x 5) (3x+2) = Multiply 3 × -10 = Find the two numbers that give a product of -30 and a sum of -13. = 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 to get 3x (x − 5) + 2(x 5) = Replace the middle term with -15x and 2x.
Answers
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GPT-4o mini
To factor the expression \(3x^2 - 13x - 10\) step by step, we would follow the order of operations provided and clarify some points along the way. Here’s the correct sequence to achieve the desired result:
1. **Determine if there is a greatest common factor amongst all the terms.**
(In this case, there isn't a GCF other than 1 for the coefficients.)
2. **Multiply 3 × -10.**
(This gives us -30 which is the product of the leading coefficient and the constant term.)
3. **Find the two numbers that give a product of -30 and a sum of -13.**
(The numbers are -15 and 2 because -15 × 2 = -30 and -15 + 2 = -13.)
4. **Replace the middle term with -15x and 2x.**
(This gives us the expression: \(3x^2 - 15x + 2x - 10\).)
5. **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.**
(This results in \(3x(x - 5) + 2(x - 5)\).)
6. **Factor the common factor once more to get your final answer.**
(Now, factor out \((x - 5)\) to obtain \((x - 5)(3x + 2)\).)
7. **Final answer: (x - 5)(3x + 2).**
So the ordered steps are as follows:
1. Determine if there is a greatest common factor amongst all the terms.
2. Multiply 3 × -10.
3. Find the two numbers that give a product of -30 and a sum of -13.
4. Replace the middle term with -15x and 2x.
5. 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 to get \(3x(x - 5) + 2(x - 5)\).
6. Factor the common factor once more to get your final answer.
7. Final answer: \((x - 5)(3x + 2)\).
1. **Determine if there is a greatest common factor amongst all the terms.**
(In this case, there isn't a GCF other than 1 for the coefficients.)
2. **Multiply 3 × -10.**
(This gives us -30 which is the product of the leading coefficient and the constant term.)
3. **Find the two numbers that give a product of -30 and a sum of -13.**
(The numbers are -15 and 2 because -15 × 2 = -30 and -15 + 2 = -13.)
4. **Replace the middle term with -15x and 2x.**
(This gives us the expression: \(3x^2 - 15x + 2x - 10\).)
5. **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.**
(This results in \(3x(x - 5) + 2(x - 5)\).)
6. **Factor the common factor once more to get your final answer.**
(Now, factor out \((x - 5)\) to obtain \((x - 5)(3x + 2)\).)
7. **Final answer: (x - 5)(3x + 2).**
So the ordered steps are as follows:
1. Determine if there is a greatest common factor amongst all the terms.
2. Multiply 3 × -10.
3. Find the two numbers that give a product of -30 and a sum of -13.
4. Replace the middle term with -15x and 2x.
5. 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 to get \(3x(x - 5) + 2(x - 5)\).
6. Factor the common factor once more to get your final answer.
7. Final answer: \((x - 5)(3x + 2)\).
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