I am having a real problem trying to figure out how to factor polynomials. I would love a thorough explanation on these questions if possible =D1) Factor each polynomial (a) f(x)= 6x^3 - x^2 - 17x...

I am having a real problem trying to figure out how to factor polynomials. I would love a thorough explanation on these questions if possible =D

1) Factor each polynomial

(a) f(x)= 6x^3 - x^2 - 17x - 10

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lfryerda | High School Teacher | (Level 2) Educator

Posted on

If you have more than one question you need to make separate posts.

To factor this polynomial, we need to examine the factors of the leading coefficient (6 in this case) and the constant coefficient (-10 in this case).

Any rational factors of the polynomial must have the numerator as a factor of -10 and the denominator as a factor of 6.  Then we guess which factors to pick.  The ones that work will have the function evaluate to zero.

Trying x=-1, we get:

`f(-1)=6(-1)^3-(-1)^2-17(-1)-10`

`=-6-1+17-10=0`

so x+1 is a factor of the polynomial.

Now using synthetic division, we can find the remaining factors.

-1 |   6   -1   -17   -10

    |        -6     7      10

    --------------------------

         6   -7    -10    0

so the remaining quadratic is `6x^2-7x-10` .  This can be factored using decomposition:

`6x^2-7x-10`    find two numbers that mutiply to 6x(-10)=-60 and add to -7

`=6x^2-12x+5x-10`   factor by grouping

`=6x(x-2)+5(x-2)`

`=(6x+5)(x-2)`

The polynomial in factored form is given by `f(x)=(x+1)(6x+5)(x-2)` .

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