Using only algebra, find a cubic function with the given zeros.
-2, 3, -7
the polynomial function is: f(x)=x^3+ (BLANK) x^2-13x-42
i can't figure out what BLANK is
Thank you!
(x+2)(x-3)(x+7) has those zeros
(x+2)(x^2 + 4x - 21)
= x*(x^2 + 4x - 21) +2*(x^2 + 4x - 21)
= x^3 + 4x^2 - 21 x + 2 x^2 + 8 x -42
= x^3 + 6 x^2 - 13 x - 42
if r is a root (zero) , then (x-r) is a factor
f(x) = (x + 2)(x - 3)(x + 7) ... multiply the binomials
To find the cubic function with the given zeros, we can use the fact that if a polynomial has a zero at a certain value, then (x - zero) is a factor of the polynomial.
Given the zeros -2, 3, and -7, we can set up the corresponding factors:
(x - (-2))(x - 3)(x - (-7)) = 0
Simplifying this expression, we get:
(x + 2)(x - 3)(x + 7) = 0
Expanding the equation, we have:
(x^2 - x - 6)(x + 7) = 0
Now, we can multiply the two binomials to get the cubic function:
f(x) = (x^2 - x - 6)(x + 7)
To find the coefficient of the x^2 term, you can either multiply the two binomials using the FOIL method or long multiplication. After multiplying, you will find that the coefficient of the x^2 term is 1.
Therefore, the polynomial function is:
f(x) = x^3 + x^2 - 13x - 42
To find the missing coefficient, we need to determine the factors of the cubic function using the given zeros.
Since -2, 3, and -7 are zeros of the function, we can write three separate linear equations as follows:
(x + 2) = 0
(x - 3) = 0
(x + 7) = 0
To obtain the cubic function, we need to multiply these factors together.
(x + 2)(x - 3)(x + 7) = 0
By multiplying these factors out, we get:
(x^2 - x - 6)(x + 7) = 0
Expanding this further, we have:
(x^3 + 7x^2 - x^2 - 7x - 6x - 42) = 0
Combining like terms, we simplify to:
x^3 + 6x^2 - 13x - 42 = 0
Therefore, the missing coefficient that replaces the (BLANK) in the equation f(x) = x^3 + (BLANK)x^2 - 13x - 42 is 6.
The final polynomial function is:
f(x) = x^3 + 6x^2 - 13x - 42.