๐ A polynomial function has the form:
f(x) = aโxโฟ + aโโโxโฟโปยน + ... + aโx + aโ
๐ฑ ROOTS (Zeros) of a Polynomial:
The roots are the x-values where f(x) = 0. To find them, factor the polynomial.
โก KEY STEPS to find roots:
1๏ธโฃ Factor out any common GCF first
2๏ธโฃ Factor the remaining polynomial
3๏ธโฃ Set each factor = 0 and solve for x
๐ MULTIPLICITY: The exponent on each factor tells you the multiplicity.
If f(x) = (x - r)แต ยท (other factors), then r is a root with multiplicity m
๐ Multiplicity Behavior at x-intercepts:
โข Odd multiplicity (1, 3, 5...) โ graph CROSSES through the x-axis โ๏ธ
โข Even multiplicity (2, 4, 6...) โ graph TOUCHES (bounces off) the x-axis ๐
๐ฏ X-INTERCEPTS vs NOT: A root IS an x-intercept if it's a real number. Complex roots (imaginary numbers like โ(-1)) are NOT x-intercepts.
โ
Example 1 โ Finding Roots & Multiplicity
Given: f(x) = 3xโด โ 6xยณ โ 9xยฒ
Step 1 (GCF): f(x) = 3xยฒ(xยฒ โ 2x โ 3)
Step 2 (Factor): f(x) = 3xยฒ(x โ 3)(x + 1)
Step 3 (Roots): x = 0 (mult. 2), x = 3 (mult. 1), x = โ1 (mult. 1)
โ All are real โ All ARE x-intercepts!
๐ At x = 0: even multiplicity โ TOUCHES (bounces)
โ๏ธ At x = 3 and x = โ1: odd multiplicity โ CROSSES
โ
Example 2 โ End Behavior
Leading term determines end behavior:
โข Even degree, positive leading coeff โ both ends go UP โโ
โข Even degree, negative leading coeff โ both ends go DOWN โโ
โข Odd degree, positive leading coeff โ left DOWN โ, right UP โ
โข Odd degree, negative leading coeff โ left UP โ, right DOWN โ
โ
Example 3 โ Y-Intercept
Y-intercept: plug in x = 0
f(x) = 2xยณ โ 5xยฒ + 3 โ f(0) = 3 โ y-intercept is (0, 3)