Answer:
Step-by-step explanation:
Given : [tex]p(x)=x^{3} +6x^{2} -7x-60[/tex]
Solution :
Part A:
First find the potential roots of p(x) using rational root theorem;
So, [tex]\text{Possible roots} =\pm\frac{\text{factors of constant term}}{\text{factors of leading coefficient}}[/tex]
Since constant term = -60
Leading coefficient = 1
[tex]\text{Possible roots} =\pm\frac{\text{factors of 60}}{\text{factors of 1}}[/tex]
[tex]\text{Possible roots} =\pm\frac{1,2,3,4,5,6,10,12,15,20,60}{1}[/tex]
Thus the possible roots are [tex] \pm1, \pm2, \pm 3, \pm4, \pm5,\pm6, \pm10, \pm12, \pm15, \pm20, \pm60[/tex]
Thus from the given options the correct answers are -10,-5,3,15
Now For Part B we will use synthetic division
Out of the possible roots we will use the root which gives remainder 0 in synthetic division :
Since we can see in the figure With -5 we are getting 0 remainder.
Refer the attached figure
We have completed the table and have obtained the following resulting coefficients: 1 , 1,−12,0. All the coefficients except the last one are the coefficients of the quotient, the last coefficient is the remainder.
Thus the quotient is [tex]x^{2} +x-12[/tex]
And remainder is 0 .
So to get the other two factors of the given polynomial we will solve the quotient by middle term splitting
[tex]x^{2} +x-12=0[/tex]
[tex]x^{2} +4x-3x-12=0[/tex]
[tex]x(x+4)-3(x+4)=0[/tex]
[tex](x-3)(x+4)=0[/tex]
Thus x-3 and x+4 are the other two factors
So , p(x)=(x+5)(x-3)(x+4)
