1.without using a calculator, find all the roots of the equation.a. x^3+x^2+4x+4=0b. x^3+4x^2+x-6=0c. x^4-4x^3+x^2+12x-12=0 2.find all the zeros of the function a. y=x^4-6x^2+8b. y=x^3+6x^2+x+6c. g(x)=x^3-4x^2-x+22d. y=x^4-x^3-5x^2-x-6
Using rational roots theorem, we can see that the possible roots of the the given equation are: [tex]\pm1,\ \pm2,\ \pm3,\ \pm6[/tex]
By substituting the possible roots, we can see that x = 1 is a root, thus x - 1 is a factor. We can get the other factors by using sythetic division to divide [tex]x^3+4x^2+x-6[/tex] by x - 1.
1 | 1 4 1 -6 | | 1 5 6 |_____________ 1 5 6 0
Thus the other factor is [tex]x^2+5x+6=(x+2)(x+3)[/tex]
Therefore, all the roots of [tex]x^3+4x^2+x-6=0[/tex] are x = 1, x = -2 and x = -3.
Part 1C:
Given
[tex]x^4-4x^3+x^2+12x-12=0[/tex]
Using rational roots theorem, we can see that the possible roots of the the given equation are: [tex]\pm1,\ \pm2,\ \pm3,\ \pm4,\ \pm6,\ \pm12[/tex]
By substituting the possible roots, we can see that x = 2 is a root, thus x - 2 is a factor. We can get the other factors by using sythetic division to divide [tex]x^4-4x^3+x^2+12x-12[/tex] by x - 2.
Using rational zeros theorem, we can see that the possible zeros of the
the given equation are: [tex]\pm1,\ \pm2,\ \pm11,\ \pm22[/tex]
By substituting the possible zeros, we can see that x = -2 is a zero, thus x + 2 is a factor. We can get the other factors by using sythetic division to divide [tex]x^3-4x^2-x+22[/tex] by x + 2.
Therefore, all the zeros of [tex]g(x)=x^3-4x^2-x+22[/tex] are x = -2, [tex]x=3\pm i\sqrt{2}[/tex].
Part 2D:
Given
[tex]y=x^4-x^3-5x^2-x-6[/tex]
Using rational zeros theorem, we can see that the possible zeros of the
the given equation are: [tex]\pm1,\ \pm2,\ \pm3,\ \pm6[/tex]
By substituting the possible zeros, we can see that x = -2 is a zero, thus x + 2 is a factor. We can get the other factors by using sythetic division to divide [tex]x^4-x^3-5x^2-x-6[/tex] by x + 2.