Answers:
- k = 23
- k = 10
- k = 5
- k = 2
- k = -2
- k = -5
- k = -10
- k = -23
There are eight items in that list above.
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Step-by-step explanation:
We'll use the AC method of factoring.
The first term has coefficient 3. The last term is -8. Multiply them to get 3*(-8) = -24.
Now let's list all the ways to multiply two integers to get -24
- -1*24
- -2*12
- -3*8
- -4*6
- -6*4
- -8*3
- -12*2
- -24*1
For each pair of factors listed, we'll add up those pairs. Each sum we get represents a possible value of k
- k = -1+24 = 23
- k = -2+12 = 10
- k = -3+8 = 5
- k = -4+6 = 2
- k = -6+4 = -2
- k = -8+3 = -5
- k = -12+2 = -10
- k = -24+1 = -23
Note the symmetry. For instance we have 10 as one sum and -10 as another sum.
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Let's go over an example. Let's say that k = 10
The expression 3x^2+kx-8 turns into 3x^2+10x-8
From the list above, we see that -2+12 = 10. So the two values -2 and 12 will help us factor.
Break up the 10x into -2x+12x and then factor by grouping like so
3x^2 + 10x - 8
3x^2 - 2x + 12x - 8
(3x^2 - 2x) + (12x - 8)
x(3x-2) + (12x - 8)
x(3x-2) + 4(3x - 2)
(x+4)(3x-2)
Therefore, 3x^2 + 10x - 8 factors into (x+4)(3x-2)
We can use the FOIL rule, distribution, or the box method to confirm this.
I'll let you try out other values of k to see how 3x^2+kx-8 factors.