a If (x-a) is a factor of a polynomial function then x=a is one of its zeroes.
B
b If (x-a) is a factor of a polynomial function then x=a is one of its zeroes.
A
aExample Solution: P(x)=x^3-6x^2+12x -8
B
bExample Solution: P(x)=x^3+6x^2+12x +8
Practice makes perfect
a We can have a polynomial with just positive roots. There is not restriction for that. Recall that we can construct a polynomial function with specific zeroes using binomial factors.
p(x) = (x-x_1)(x-x_2)... (x-x_n)
This would give us a n degree polynomial function with x_1, x_2, ... x_n as zeroes. Then, we can construct our counter example by choosing just positive zeroes to obtain a degree 3 polynomial. For example, P(x)=(x-2)^3 will have all of its 3 zeroes at x=2. We can expand it to present it in its standard form.
Notice that this is just an example solution, and there are infinitely many polynomials satisfying the given conditions.
b We can have a polynomial with just negative roots. There is not restriction for that. We can construct a polynomial function with specific zeroes using binomial factors.
p(x) = (x-x_1)(x-x_2)... (x-x_n)
This would give us a n degree polynomial function with x_1, x_2, ... x_n as zeroes. Then, we can construct our counter example choosing just positive zeroes to obtain a degree 3 polynomial. For example, P(x)=(x-(- 2) )^3 will have all of its 3 zeroes at x=- 2. We can expand it to present it in its standard form.
