What is the multiplicity of the roots of y = (x – 1)4(x – 3)7(x + 2)1?
Hint
Check the exponents of each factor.
Answer
x = 1 has multiplicity 4, x = 3 has multiplicity 7, and x = -2 has multiplicity 1.
Example 2
y = x4 − 6x3 + 13x2 – 12x + 4 has zeros at 2 and 1. What is their multiplicity?
Hint
One round of synthetic division by x – 2 results in y = (x – 2)(x3 − 5x2 + 8x – 4). Try to find the factorization of the quotient by x – 1.
Answer
They each have multiplicity 2.
Example 3
Will f(x) = (x − 21)4(x − 6) go from positive to negative at x = 21 or keep the same sign?
Hint
It has even multiplicity.
Answer
The sign will stay the same, as it will tap the axis at x = 21.
Example 4
Will f(x) = -(x − 9)3(x − 6) go from negative to positive at x = 6 or keep the same sign?
Hint
It has odd multiplicity, so check the signs of each factor on each side of 6. Also, don't forget the "–" sign out in front.
Answer
It goes from negative to positive.
Example 5
A polynomial is even with a positive leading coefficient. Its only real zeros are of even multiplicity. Could the polynomial ever have a y-value of -3?
Hint
The polynomial has both arms up, due to its leading coeffient.