Factor using the Factor Theorem: `6x^4 -19x^3 -2x^2+44x-24`

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tiburtius | High School Teacher | (Level 2) Educator

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If there is a solution `x_0` of algebraic equation

`a_nx^x+a_(n-1)x^(n-1)+cdots+a_1x+a_0=0`                                

` `that is a whole number than that solution must divide `a_0` i.e. `x_0|a_0`. Also if `x_0` is such solution then polynomial ` ` `a_nx^x+a_(n-1)x^(n-1)+cdots+a_1x+a_0` can be divided by `x-x_0` without residue.

Now if you put `x_0=2` you see that it is a null point of your polynomial

`6cdot2^4-19cdot2^3-2cdot2^2+44cdot2-24=96-152-8+88-24=0`

`(6x^4-19x^3-2x^2+44x-24):(x-2)=6x^3-7x^2-16x+12`

Again we search for integer solution and we get `x_1=2`

`6cdot2^3-7cdot2^2-16cdot2+12=48-28-32+12=0`

`(6x^3-7x^2-16x+12) :(x-2)=6x^2+5x-6`

Now this is a quadratic polynomial for which we can easily find null points.

`x_(2,3)=(-5 pm sqrt(25+144))/(2cdot6)=(-5 pm 13)/12`

`x_2=-3/2`, `x_3=2/3`

Now that we have all null points we can use them to factor our polynomial

`6x^4-19x^3-2x^2+44x-24=(x-x_0)(x-x_1)(x-x_2)(x-x_3)=`

`(x-2)^2(x+3/2)(x-2/3)`  <-- Your solution.

 

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