Respuesta :
The polynomial is given to be:
[tex]f(x)=-2x^2+14x+120[/tex]FACTORING THE POLYNOMIAL
Step 1: Multiply the first and last term of the polynomial and get two numbers that will multiply to give the result and will add up to the middle term
[tex]\begin{gathered} -2x^2\times120=-240x^2 \\ Numbers=-10x,+24x \end{gathered}[/tex]Step 2: Replace the middle term with the two numbers gotten in Step 1 above
[tex]f(x)=-2x^2-10x+24x+120[/tex]Step 3: Factor out the common term in each pair of numbers as shown below
[tex]\begin{gathered} f(x)=(-2x^2-10x)+(24x+120) \\ f(x)=-2x(x+5)+24(x+5) \end{gathered}[/tex]Step 4: Factor out the common term (x + 5)
[tex]f(x)=(x+5)(-2x+24)[/tex]Step 5: Factor out -2x from the term (-2x + 24)
[tex]f(x)=-2(x+5)(x-12)[/tex]The factored polynomial is:
[tex]f(x)=-2(x+5)(x-12)[/tex]ZEROES OF THE FUNCTION
The zeroes of the function are gotten at f(x) = 0
[tex]f(x)=0[/tex]Therefore, we have that:
[tex]-2(x+5)(x-12)=0[/tex]Recall the Zero Factor Principle:
[tex]\begin{gathered} \text{If} \\ ab=0 \\ \text{then} \\ a=0,b=0 \end{gathered}[/tex]Therefore, we have that:
[tex]\begin{gathered} x+5=0 \\ \therefore \\ x=-5 \end{gathered}[/tex]or
[tex]\begin{gathered} x-12=0 \\ \therefore \\ x=12 \end{gathered}[/tex]Therefore, the zeroes of the function are:
[tex]x=-5\text{ }or\text{ }x=12[/tex]