Check my answer please? Not sure if I did this right.
Let y = f(x) be a cubic polynomial with leading coefficient a = 2 and f(-2) = f(1) = f(2) = 0. find the factored form of f.
I got: f(x) = 2(x + 2)(x - 1)(x - 2)
Hmm, let me see.
f(-2)=0, check
f(1)=0, check
f(2)=0, check
when expanded, the leading coefficient is 2
(polynomial=2x³-2x²-8x+8)
This checks too.
Yes, your answer is correct.
Well, let me put on my math hat and check your answer for you.
Good news! Your answer is absolutely correct! Give yourself a pat on the back. You've correctly factored the cubic polynomial with the given information. You must have a real talent for math, or at least a good sense of humor! Keep up the great work!
To check your answer, we need to verify if the given polynomial satisfies the given conditions.
Given that the polynomial is cubic and has a leading coefficient a = 2, the factored form of the polynomial should be in the form:
f(x) = a(x - r1)(x - r2)(x - r3)
where r1, r2, and r3 are the roots of the polynomial.
Given that f(-2) = f(1) = f(2) = 0, it means that the roots of the polynomial are -2, 1, and 2.
Therefore, plugging in these values into the factored form, we have:
f(x) = 2(x - (-2))(x - 1)(x - 2)
= 2(x + 2)(x - 1)(x - 2)
Based on the given conditions and your answer, f(x) = 2(x + 2)(x - 1)(x - 2) is indeed the correct factored form of the cubic polynomial with the given conditions. Well done!
To check if your answer is correct, we can use the given conditions and see if the factored form you obtained satisfies those conditions.
Given conditions:
1. Leading coefficient: a = 2
2. f(-2) = f(1) = f(2) = 0
Let's substitute the values into the factored form you obtained: f(x) = 2(x + 2)(x - 1)(x - 2)
1. f(-2):
When x = -2, we have f(-2) = 2(-2 + 2)(-2 - 1)(-2 - 2) = 2(0)(-3)(-4) = 0
So, the condition f(-2) = 0 is satisfied.
2. f(1):
When x = 1, we have f(1) = 2(1 + 2)(1 - 1)(1 - 2) = 2(3)(0)(-1) = 0
So, the condition f(1) = 0 is satisfied.
3. f(2):
When x = 2, we have f(2) = 2(2 + 2)(2 - 1)(2 - 2) = 2(4)(1)(0) = 0
So, the condition f(2) = 0 is satisfied.
Since the factored form you obtained satisfies all the given conditions, your answer is correct:
f(x) = 2(x + 2)(x - 1)(x - 2)