Question

A polynomial function has a root of –3 with multiplicity 2, a root of 0 with multiplicity 1, a root of 1 with multiplicity 1, and a root of 3 with multiplicity 2. If the function has a positive leading coefficient and is of even degree, which could be the graph of the function?

Answer 1

Finding polynomial.

Let the required polynomial be f (x).

• 1. A root (- 3) with multiplicity 2
• Then (x + 3) (x + 3) is a factor of f (x).

• 2. A root 0 with multiplicity 1
• Then x is a factor of f (x).

• 3. A root of 1 with multiplicity 1
• Then (x - 1) is a factor of f (x).

• 4. A root of 3 with multiplicity 2
• Then (x - 3) (x - 3) is a factor of f (x).

Therefore the polynomial f (x) is given by

f (x) = x (x + 3) (x + 3) (x - 1) (x - 3) (x - 3)

= x (x - 1) (x + 3) (x - 3) (x + 3) (x - 3)

= x (x - 1) (x² - 9) (x² - 9)

= (x² - x) (x⁴ - 18x² + 81)

= x⁶ - 18x⁴ + 81x² - x⁵ + 18x³ - 81x

= x⁶ - x⁵ - 18x⁴ + 18x³ + 81x² - 81x

[[ Refer to the given attachment for its graph. ]]