Answer :
To factor the polynomial [tex]\( x^4 + 2x^3 - 7x^2 - 8x + 12 \)[/tex] with the given factor [tex]\((x - 2)\)[/tex]:
1. Polynomial Division:
We start by dividing the polynomial [tex]\( x^4 + 2x^3 - 7x^2 - 8x + 12 \)[/tex] by [tex]\( x - 2 \)[/tex].
The quotient from this division is [tex]\( x^3 + 4x^2 + x - 6 \)[/tex].
2. Factoring the Quotient Polynomial:
Now we need to factor [tex]\( x^3 + 4x^2 + x - 6 \)[/tex].
3. Finding the Factors:
After factoring [tex]\( x^3 + 4x^2 + x - 6 \)[/tex], we get the factors [tex]\( (x - 1) \)[/tex], [tex]\( (x + 2) \)[/tex], and [tex]\( (x + 3) \)[/tex].
So, the other factors of the polynomial [tex]\( x^4 + 2x^3 - 7x^2 - 8x + 12 \)[/tex] besides [tex]\( (x - 2) \)[/tex] are:
[Choose [tex]\( x + 2 \)[/tex]],
[Choose [tex]\( x + 3 \)[/tex]], and
[Choose [tex]\( x - 1 \)[/tex]].
1. Polynomial Division:
We start by dividing the polynomial [tex]\( x^4 + 2x^3 - 7x^2 - 8x + 12 \)[/tex] by [tex]\( x - 2 \)[/tex].
The quotient from this division is [tex]\( x^3 + 4x^2 + x - 6 \)[/tex].
2. Factoring the Quotient Polynomial:
Now we need to factor [tex]\( x^3 + 4x^2 + x - 6 \)[/tex].
3. Finding the Factors:
After factoring [tex]\( x^3 + 4x^2 + x - 6 \)[/tex], we get the factors [tex]\( (x - 1) \)[/tex], [tex]\( (x + 2) \)[/tex], and [tex]\( (x + 3) \)[/tex].
So, the other factors of the polynomial [tex]\( x^4 + 2x^3 - 7x^2 - 8x + 12 \)[/tex] besides [tex]\( (x - 2) \)[/tex] are:
[Choose [tex]\( x + 2 \)[/tex]],
[Choose [tex]\( x + 3 \)[/tex]], and
[Choose [tex]\( x - 1 \)[/tex]].
