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:
-
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.) -
Multiply 3 × -10.
(This gives us -30 which is the product of the leading coefficient and the constant term.) -
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.) -
Replace the middle term with -15x and 2x.
(This gives us the expression: \(3x^2 - 15x + 2x - 10\).) -
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.
(This results in \(3x(x - 5) + 2(x - 5)\).) -
Factor the common factor once more to get your final answer.
(Now, factor out \((x - 5)\) to obtain \((x - 5)(3x + 2)\).) -
Final answer: (x - 5)(3x + 2).
So the ordered steps are as follows:
- Determine if there is a greatest common factor amongst all the terms.
- Multiply 3 × -10.
- Find the two numbers that give a product of -30 and a sum of -13.
- Replace the middle term with -15x and 2x.
- 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 \(3x(x - 5) + 2(x - 5)\).
- Factor the common factor once more to get your final answer.
- Final answer: \((x - 5)(3x + 2)\).
1 answer