Order the steps from first to last that would be used to factor:

Question

3x^2 - 13x - 10

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) + (x-5),
Replace the middle term with -15x and 2x,
Multiply 3 x -10,
Factor the common factor once more to get your final answer,
Find the two numbers that give a product of -30 and a sum of -13,
Determine if there is a greatest common factor amongst all the terms,
Final answer: (x-5) (3x+2)

Answer 1

To factor the expression \(3x^2 - 13x - 10\) using the method of grouping, we can order the steps as follows:

1. Determine if there is a greatest common factor amongst all the terms.

   • In this case, there is no greatest common factor.

2. Find the two numbers that give a product of -30 and a sum of -13.

   • The two numbers are -15 and 2 (\(-15 \times 2 = -30\) and \(-15 + 2 = -13\)).

3. Replace the middle term with -15x and 2x.

   • Rewrite the expression as \(3x^2 - 15x + 2x - 10\).

4. 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 gives us \(3x(x - 5) + 2(x - 5)\).

5. Factor the common factor once more to get your final answer.

   • Combine the groups to get \((x - 5)(3x + 2)\).

6. Final answer: \((x - 5)(3x + 2)\).

So the ordered steps are:

1. Determine if there is a greatest common factor amongst all the terms.
2. Find the two numbers that give a product of -30 and a sum of -13.
3. Replace the middle term with -15x and 2x.
4. 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.
5. Factor the common factor once more to get your final answer.
6. Final answer: \((x - 5)(3x + 2)\).