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Math Help - how to factor---x^3+x^2-17x+15?????

  1. #1
    Member sluggerbroth's Avatar
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    how to factor---x^3+x^2-17x+15?????

    book says (x+5)(x-3)(x-1)

    Help! I do not know procedure to factor trinomial??????????
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  2. #2
    Member Sylvia104's Avatar
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    Re: How to factor x^3+x^2-17x+15?

    Let f(x)=x^3+x^2-17x+15.

    Try a number a and see if f(a)=0, say a=1.

    f(1) = 1^3+1^2-17(1)+15 = 0.

    Thus x-1 is a factor, i.e. x^3+x^2-17x+15=(x-1)g(x) where g(x) is a quadratic polynomial. Once you've found g(x) you can factorize it further.
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  3. #3
    Member sluggerbroth's Avatar
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    Re: How to factor x^3+x^2-17x+15?

    keep talking because i need more explanation????
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    Re: how to factor---x^3+x^2-17x+15?????

    To "factor" a polynomial with integer coefficients typically means "write as a product of factors with integer coefficients. For example, x^2- 3 can be written as (x- \sqrt{3})(x+ \sqrt{3}) but we don't normally think of that as "factoring".

    What Sylvia104 did was look for rational zeros of the polynomial. There is a "rational root theorem" that says "Any rational number roots of the polynomial equation, a_nx^n+ a_{n-1}x^{n-1}+ \cdot\cdot\cdot+ a_1x+ a_0= 0, are of the form \frac{m}{n} with m an integer that evenly divides a_0 and n an integer that evenly divides a_n. Here, a_n= 1 so any rational root must have denominator 1- in other words be an integer. a_0= 15 so the only possible rational roots are 1, -1, 3, -3, 5, -5, 15, and -15, the factors of 15.
    Check each one: [tex](1)^3+ (1)^2- 17(1)+ 15= 1+ 1- 17+ 15= 0. Yes! 1 is a root so x- 1 is a factor.
    [tex](-1)^3+ (-1)^2- 17(-1)+ 15= -1+ 1+ 17+ 15= 32, not 0. -1 is not a root so x-(-1)= x+ 1 is not a factor.
    (3)^3+ (3)^2- 17(3)+ 15= 27+ 9- 51+ 15= 0 so 3 is a root and x- 3 is a factor.
    (-3)^3+ (-3)^2- 17(-3)+ 15= -27+ 9+ 51+ 15= 48 so -3 is not a root and x-(-3)= x+ 3 is not a factor.
    (5)^3+ (5)^2- 17(5)+ 15= 125+ 25- 85+ 15= 80 so 5 is not a root and x- 5 is not a factor.
    (-5)^3+ (-5)^2- 17(-5)+ 15= -125+ 25+ 85+ 15= 0 so -5 is a root and x- (-5)= x+ 5 is a factor.

    Having found three factors, we don't need to check 15 and -15 as roots- we now know that
    x^3+ x^2- 17x+ 15= (x+5)(x-3)(x-1)

    I suspect that Sylvia104's point was that having seen that 1+ 1- 17+ 15= 0 so that x= 1 is a zero of the polynomial and x- 1 a factor, you could divide x^3+ x^2- 17x+ 15 by x- 1, leaving x^2+ 2x- 15 which is relatively easy to as (x- 3)(x+ 5).
    Thanks from sluggerbroth
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