If f(-3)=0 determine the factors of f(x)=8x^3+74x^2+200x+150
haven't been able to figure out this question, the assignments due tonight if anyone could help that would be amazing
To determine the factors of f(x) = 8x^3 + 74x^2 + 200x + 150, we can use the given information that f(-3) = 0.
To find the factors, we need to find the values of x that make f(x) equal to zero. If f(-3) = 0, it means that substituting x = -3 into the equation results in f(-3) equal to zero.
Let's substitute x = -3 into the equation:
f(-3) = 8(-3)^3 + 74(-3)^2 + 200(-3) + 150
= 8(-27) + 74(9) - 600 + 150
= -216 + 666 - 600 + 150
= 0
We have verified that f(-3) equals zero, which means (x + 3) is a factor of f(x). This is known as the factor theorem.
To determine the other possible factors, we can divide f(x) by (x + 3) to obtain the remaining quotient.
Using polynomial long division or synthetic division, divide f(x) by (x + 3):
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x + 3 | 8x^3 + 74x^2 + 200x + 150
The division will yield a quotient and a remainder. If the remainder is zero, then the divisor (x + 3) is a factor of f(x).
Once the division is completed, the quotient will be a quadratic or linear polynomial, depending on the original degree of the cubic polynomial f(x). We can then factor this resulting polynomial further if necessary.
By repeating this process, we can determine all the factors of f(x).
Review the Remainder Theorem. If f(a) = 0 then (x-a) is a factor of f(x)
A little synthetic division shows that
f(x) = (x+3)(8x^2+50x+50)
Now just factor the quadratic as usual