When you have a bunch like this, I find it easiest to graph them.
The numbers of real solutions are ... 2 0 2 1
I know you know how to use Desmos to graph these. You can also write a little spreadsheet program to compute the discriminant b^2 -4ac, or do it the hard way. 1^2 -4(-3)(12) = 145 . . . positive, so 2 solutions (-6)^2 -4(2)(5) = -4 . . .. negative, so 0 solutions 7^2 -4(1)(-11) = 93 . . . .. positive, so 2 solutions (-8)^2 -4(-1)(-16) = 0 . .. zero, so 1 solution
When you have a bunch like this, I find it easiest to graph them.
The numbers of real solutions are ... 2 0 2 1
I know you know how to use Desmos to graph these. You can also write a little spreadsheet program to compute the discriminant b^2 -4ac, or do it the hard way. 1^2 -4(-3)(12) = 145 . . . positive, so 2 solutions (-6)^2 -4(2)(5) = -4 . . .. negative, so 0 solutions 7^2 -4(1)(-11) = 93 . . . .. positive, so 2 solutions (-8)^2 -4(-1)(-16) = 0 . .. zero, so 1 solution
Each of the latter three answers demonstrates one change of sign in its coefficients. According to Descarte's Rule of Signs, that means each of them has one positive real root. If there is a positive real root, the only two real roots will not be -4 and -1 (both negative).
The only viable option is the first one.
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There are other ways you can go at this. I like to let a graphing calculator show the zeros of functions of higher degree. Only the first function has only the two real roots given.
You can also evaluate each function or do synthetic division to see which polynomial has -1 or -4 as a root. Shown in the second attachment is the first function evaluated for x=-1.
Each of the latter three answers demonstrates one change of sign in its coefficients. According to Descarte's Rule of Signs, that means each of them has one positive real root. If there is a positive real root, the only two real roots will not be -4 and -1 (both negative).
The only viable option is the first one.
_____
There are other ways you can go at this. I like to let a graphing calculator show the zeros of functions of higher degree. Only the first function has only the two real roots given.
You can also evaluate each function or do synthetic division to see which polynomial has -1 or -4 as a root. Shown in the second attachment is the first function evaluated for x=-1.
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