We want to completely factor the given expression. To do so, we will first identify and factor out the greatest common factor.
Factor Out the GCF
The greatest common factor (GCF) of an expression is a common factor of the terms in the expression. It is the common factor with the greatest coefficient and the greatest exponent. The GCF of the given expression is 2t.
Here we have a quadratic trinomial of the form ax^2+bx+c, where |a| ≠1 and there are no common factors. To factor this expression, we will rewrite the middle term, bx, as two terms. The coefficients of these two terms will be factors of ac whose sum must be b.
2t( 2t^2+5t-42 ) ⇔ 2t( 2t^2+ 5t+( -42) )
We have that a= 2, b= 5, and c= -42. There are now three steps we need to follow in order to rewrite the above expression.
Find a c. Since we have that a= 2 and c= -42, the value of a c is 2* -42=- 84.
Find factors of a c. Since a c=- 42, which is negative, we need factors of a c to have opposite signs — one positive and one negative — in order for the product to be negative. Since b= 5, which is positive, the absolute value of the positive factor will need to be greater than the absolute value of the negative factor, so that their sum is positive.
Factor Pair
Product
Sum
-1 and 84
-1* 84 - 84
-1+84 83
- 2 and 42
- 2* 42 -84
- 2+42 40
-3 and 28
-3* 28 -84
-3+28 25
- 4 and 21
- 4* 21 -84
- 4+21 17
-6 and 14
-6* 14 - 84
-6+14 8
- 7 and 12
- 7* 12 - 84
- 7+12 5
Rewrite bx as two terms. Now that we know which factors are the ones to be used, we can rewrite bx as two terms.
2t(2t^2+ 5t-42 )
⇕
2 ( 2x^2 + 12t - 7t-42 )
Finally, we will factor the last expression obtained.
We can see above that after expanding and simplifying, the result is the same as the given expression. Therefore, we can be sure our solution is correct!
