I need help in understanding these operations factor by grouping 8x^3-56^2-5x+35
Is the second term supposed to be 56x^2? If so, then use the following as an example to factor by grouping
x^3 - 3x^2 + 2x - 6
= (x^3 - 3x^2) + (2x - 6). Now factor out the common factors from each group such as x^2(x - 3) + 2(x - 3). That can be rewritten as (x-3)*(x^2 + 2)
In your case,
8x^3-56^2-5x+35 = 8x^2(x-7)-5(x-7)
= (8x^2-5)(x-7)
The roots are x=7 and x = +/-sqrt(5/8)
This cubic equation was "set up" to work. In by far most cases, with random integer coefficients, it doesn't.
To factor the expression 8x^3 - 56x^2 - 5x + 35 by grouping, follow these steps:
Step 1: Group the terms in pairs.
Group the first two terms, 8x^3 and -56x^2, together, and the last two terms, -5x and 35, together.
(8x^3 - 56x^2) - (5x - 35)
Step 2: Factor out the greatest common factor from each group.
In the first group, both terms have a common factor of 8x^2. When you factor it out, you get:
8x^2(x - 7)
In the second group, both terms have a common factor of -5. When you factor it out, you get:
-5(x - 7)
Now the expression becomes:
8x^2(x - 7) - 5(x - 7)
Step 3: Factor out the common binomial.
Notice that both terms now have a common factor of (x - 7). Factor it out, and you get:
(x - 7)(8x^2 - 5)
Finally, the fully factored expression is:
(x - 7)(8x^2 - 5)
That's it! You have successfully factored the expression by grouping.