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Question 1123899: Please help me solve: if f(x)=6x^3+13x^2+2x-5 and f(-1)=0, find the factors of f(x).
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if f(-1) = 0, then one of the factors has to be x+1.
that's because, if x + 1 = 0, then solving for x will get x = -1.
your equation is f(x) = 6x^3 + 13x^2 + 2x -5.
f(-1) is solved by replacing x with -1 and evaluating the expression.
f(-1) becomes 6 * (-1)^3 + 13 * (-1)^2 + 2 * (-1) - 5.
the result is -6 + 13 - 2 - 5 = 0.
therefore, you need to divide 6x^3 + 13x^2 + 2x - 5 by (x+1).
alternatively, you can use synthetic division with -1 operating on the coefficients of 6x^3 + 13x^2 + 2x - 5.
either way, you'll get 6x^3 + 13x^2 + 2x - 5 divided by (x + 1) = 6x^2 + 7x - 5.
you can factor 6x^2 + 7x - 5 and get the factors of (2x - 1) * (3x + 5)
therefore, the factors of 6x^3 + 13x^2 + 2x - 5 are (x + 1) * (2x - 1) * (3x + 5).
it can be seen graphically that these are equivalent expressions because both equations will draw the same line, as shown below.
my calculations in the long division and the synthetic division are shown below.
the top section calculations are long division.
the bottom section calculations are synthetic division.
if you don't know how to do long division or synthetic division or how to factor quadratic equations, here's some references that should help you.
https://www.purplemath.com/modules/polydiv2.htm
https://www.purplemath.com/modules/synthdiv.htm
https://www.purplemath.com/modules/factquad.htm
when factoring quadratics, if all else fails, use the quadratic formula.
that formula is:
x = -b plus or minus sqrt(b^2 - 4ac)
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2a
for example, if it tells you that x = -1, then you want that equation to be equal to 0.
you would add 1 to both sides of that equation to get x + 1 = 0.
that means (x + 1) is one of your factors.
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