Respuesta :
Solution:
The first polynomial is given below as
[tex]\begin{gathered} f(x)=x^3-3x^2-13x+15 \\ x+3=0 \\ x=-3 \\ f(-3)=(-3)^3-3(-3)^2-13(-3)+15 \\ f(-3)=-27-27+39+15 \\ f(-3)=0 \end{gathered}[/tex]Hence,
The first answer is
[tex]\begin{gathered} (x+3) \\ is\text{ a factor of} \\ f(x)=x^{3}-3x^{2}-13x+15 \end{gathered}[/tex]Step 2:
The second function is given below as
[tex]\begin{gathered} f(x)=x^4+3x^3-8x^2+5x-25 \\ x+5=0 \\ x=-5 \\ f(-5)=(-5)^4+3(-5)^3-8(-5)^2+5(-5)-25 \\ f(-5)=625-375-200-25-25 \\ f(-5)=0 \end{gathered}[/tex]Hence,
The final answer is
[tex]\begin{gathered} x+5 \\ is\text{ a factor of } \\ f(x)=x^4+3x^3-8x^2+5x-25 \end{gathered}[/tex]Step 3:
The fourth function is given below as
[tex]\begin{gathered} f(x)=x^3-2x^2-x+2 \\ x-2=0 \\ x=2 \\ f(2)=2^3-2(2^2)-2+2 \\ f(2)=8-8-2+2 \\ f(2)=0 \end{gathered}[/tex]Hence,
The final answer is
[tex]\begin{gathered} (x-2) \\ is\text{ a factor of} \\ f(x)=x^3-2x^2-x+2 \end{gathered}[/tex]Step 4:
[tex]\begin{gathered} f(x)=-x^3+13x-12 \\ x+4=0 \\ x=-4 \\ f(-4)=-(-4)^3+13(-4)-12 \\ f(-4)=64-52-12 \\ f(-4)=0 \end{gathered}[/tex]Hence,
The final answer is
[tex]\begin{gathered} x+4 \\ is\text{ a factor of } \\ f(x)=-x^3+13x-12 \end{gathered}[/tex]