Since we have been given that 4 is a zero of our equation, So we will divide our equation by (x-4) to find other factors of our equation.
Upon dividing our equation [tex]x^{3} +3x^{2} -18x-40=0[/tex] by (x-4) we will get [tex]x^{2}+7x+10=0[/tex].
Now we will factor this equation by splitting the middle term to get our other factors.
[tex]x^{2} +5x+2x+10=0[/tex]
[tex]x(x+5)+2(x+5)=0[/tex]
[tex](x+5)(x+2)=0[/tex]
We can see that our factors are (x+2)(x+5)(x-4)=0, therefore, last option is the correct choice.
The zero of an equation is the point, where the equation cross the x-axis
The factored form is: [tex]\mathbf{(x + 5)(x + 2)(x - 4) = 0}[/tex]
The function is given as:
[tex]\mathbf{f(x) = x^3 + 3x^2 - 18x - 40}[/tex]
Because 4 is a zero of the function, then we can divide both sides by x - 4
[tex]\mathbf{\frac{f(x)}{x - 4} = \frac{x^3 + 3x^2 - 18x - 40}{x - 4}}[/tex]
Expand the numerator
[tex]\mathbf{\frac{f(x)}{x - 4} = \frac{x^3 + 7x^2 -4x^2 + 10x - 28x - 40}{x - 4}}[/tex]
Factorize
[tex]\mathbf{\frac{f(x)}{x - 4} = \frac{(x^2 + 7x + 10)(x - 4)}{x - 4}}[/tex]
Cancel out common factor
[tex]\mathbf{\frac{f(x)}{x - 4} = x^2 + 7x + 10}[/tex]
Factorize
[tex]\mathbf{\frac{f(x)}{x - 4} = (x + 5)(x + 2)}[/tex]
Multiply both sides by (x - 4)
[tex]\mathbf{f(x) = (x + 5)(x + 2)(x - 4)}[/tex]
Equate to 0
[tex]\mathbf{(x + 5)(x + 2)(x - 4) = 0}[/tex]
Hence, the factored form is: [tex]\mathbf{(x + 5)(x + 2)(x - 4) = 0}[/tex]
Read more about zeros of equations at:
https://brainly.com/question/944636
