given the zeros, write a polynomial equation in factored form.zero -5,9,12 show all the step please
y = x(x+5)(x-9)(x-12)
in order for y to be zero, at least one of the factors must be zero. if k is a zero of y, then (x-k) is a factor of y.
Sure! To write a polynomial equation in factored form, we need to find the factors or zeros of the polynomial.
Given the zeros -5, 9, and 12, we can write the factors as follows:
(x + 5), (x - 9), (x - 12)
Now, to get the polynomial equation in factored form, we multiply these factors together. So the equation would be:
(x + 5)(x - 9)(x - 12)
To simplify this equation, we can multiply the factors using the distributive property. To do this, we multiply the first two factors first, and then multiply the result by the third factor. Let's go through the steps:
Step 1: Multiply the first two factors: (x + 5)(x - 9)
When using the distributive property, each term of the first factor should be multiplied by each term of the second factor.
(x + 5)(x - 9) = x(x) + x(-9) + 5(x) + 5(-9)
Simplifying this expression, we get:
= x^2 - 9x + 5x - 45
= x^2 - 4x - 45
Step 2: Multiply the result from step 1 by the third factor: (x^2 - 4x - 45)(x - 12)
Again, using the distributive property, we multiply each term of the result from step 1 by each term of the third factor.
(x^2 - 4x - 45)(x - 12) = (x^2)(x) + (x^2)(-12) + (-4x)(x) + (-4x)(-12) + (-45)(x) + (-45)(-12)
Simplifying this expression further, we get:
= x^3 - 12x^2 - 4x^2 + 48x - 45x + 540
Combining like terms, we have:
= x^3 - 16x^2 + 3x + 540
So, the polynomial equation in factored form, using the given zeros -5, 9, and 12, would be:
(x + 5)(x - 9)(x - 12) = x^3 - 16x^2 + 3x + 540