Answer/Step-by-step explanation:
Zeros are x-intercepts. (Solutions are zeros are x-intercepts!)
If you are allowed to use technology, such as a graphing calculator or an online graphing tool, input the function and look for the x-intercepts. Graphing calculators have menus to find them for you. It is a legit way to "verify the zeros" Also, since this is a third degree function (highest exponent is a 3rd power) but the leading coefficient is negative, the end behavior acts the same as:
y = -x^3
a much simpler version. Actually, even simpler:
y = -x
All of these, go up to the left and down to the right. Or some teachers/classes/books/programs want you to say:
as x goes to negative infinity, y goes to infinity (up to the left) and as x goes to infinity, y goes to negative infinity (down to the right)
SO, if you cannot use technology, put the given x-intercept into the function. And check that it gives you a value of 0. Like this:
f(x) =
-x^3-7x^2-7x+15
check by substituting -5 for x
f(-5) =
-(-5)^3-7(-5)^2-7(-5)+15
= 125 -7(25) +35+15
= 125- 175 + 35 + 15
= 0
When you use x =-5 you get y = 0. This verifies that -5 is a zero.
Check -3.
f(-3)=-(-3)^3-7(-3)^2-7(-3)+15
= 27 -7(9) +21 +15
= 27 - 63 +21 +15
= 0
Verified!
Lastly, check 1:
f(1)=-(1)^3-7(1)^2-7(1)+15
= -1 - 7 - 7 + 15
= 0
Verified!