Question

Hey guys I am struggling! Which of the following is a polynomial with roots 5, −6, and 7?

Answer 1

@AloneS

Answer 2

What's it look like?

Answer 3

Are you given choices?

Answer 4

If so, plug in each of the roots, one at a time, into each choice. The one that evaluates to 0 for all roots is the answer.

Answer 5

Ah yes, sorry.
A) f(x) = x^3 − 6x^2 − 30x + 37
B) f(x) = x^3 − 30x^2 − 37x + 42
C) f(x) = x^3 − 6x^2 − 37x + 210
D) f(x) = x^3 − 30x^2 − 42x + 210

Answer 6

If f(x) is a polynomial function, and f(a) = 0, then a is a root of f(x).
Since you want the three roots to be roots of a polynomial, you must make sure that f(5) = 0, f(-6) = 0, and f(7) = 0. You need to find for which polynomial, all those values evaluate to zero.

Answer 7

Ok, now start with A.
A) f(x) = x^3 − 6x^2 − 30x + 37
Try 5: f(5) = 5^3 − 6(5)^2 − 30(5) + 37 =
Try -6: f(-6) = (-6)^3 − 6(-6)^2 − 30(-6) + 37 =
Try 7: f(7) = 7^3 − 6(7)^2 − 30(7) + 37 =

Evaluate all the expressions above. If they are all equal to zero, that is your answer.
If not keep working through the other choices the same way.

Answer 8

Give me a sec to figure this out.

Answer 9

Not A*5

Answer 10

Second method (which may be quicker):

If a polynomial has roots a, b, c, then it has binomial factors x - a, x - b, x - c.

Set up each binomial factor using a root, 5, -6, 7, as x - 5, x - (-6), x - 7
Now multiply all these binomials together to get the polynomial.

f(x) = (x - 5)(x + 6)(x - 7)

Just multiply the right side and that will give you your polynomial.

Answer 11

I'm not sure how to multiply that. I can do two, not sure how to do 3. :\

Answer 12

@mathstudent55 Can I get instruction on that, too?

Answer 13

Never mind. :) Thank you so much! @mathstudent55

Answer 14

You multiply 2 binomials using FOIL.

Then to multiply any polynomial by any polynomial, multiply every term of the first polynomial by every term of the second polynomial. Then collect like terms.

BTW, when you FOIL two binomials, you are multiplying every term of the first binomial by every term of the second. FOIL is simply a way of doing those products in a certain order.