Solving Higher Order Equations

Discuss the role of synthetic division in solving higher order polynomials.

• Synthetic division is only applicable to quadratic polynomials

• Synthetic division simplifies the process of dividing higher order polynomials by linear factors.

• Synthetic division is a method for finding the derivative of a polynomial function

• Synthetic division is used to multiply higher order polynomials

Explain the concept of multiplicity of roots in polynomial equations.

• Multiplicity of roots is the same as the degree of the polynomial equation

• Multiplicity of roots indicates the size of the roots in the solution set

• Multiplicity of roots in polynomial equations refers to the number of variables in the equation

• In polynomial equations, multiplicity of roots refers to the number of times a particular root appears in the solution set.

Solve the following equation 0 = x³ -3x²-5x+15

• 3, √5

• 3, √5, -√5

• -3, √5, -√5

• -3, 5i, -5i

What are the solutions for the polynomial equation 0 = 3 (x + 4) (x + 1) (x - 3)?

• -4, -1, 3

• 3, -3, 4, 1

• 4, 1, -3

• 3, -4, -1, 3

What is the equation in factored form of this graph?

• f(x) = (x-4)(x-1)²(x+2)(x+4)

• f(x) = (x-4)(x+1)²(x+2)(x+4)

• f(x) = (x-4)(x-1)²(x-2)(x+4)

• f(x) = (x+4)(x+1)²(x-2)(x-4)

What is the equation of this graph in factored form?

• (x - 1)(x + 3)²

• (x - 1)²(x + 3)

• (x + 1)²(x - 3)

• (x + 1)(x - 3)²

Which of the following could be the equation of this graph in factored form? (Careful-pay attention to multiplicity.)

• f(x) = (x-4)(x-1)²(x+2)(x+4)

• f(x) = (x-4)(x+1)²(x+2)(x+4)

• f(x) = (x-4)(x-1)²(x-2)(x+4)

• f(x) = (x+4)(x+1)²(x-2)(x-4)

The polynomial
f(x) = 5x⁴ + 2x³ + 2x − 7 can have, at most, how many solutions?

• 5

• 4

• 3

• 2

What is the end-behavior of the function 
y = x⁶ - 7x⁵ + 7x - 9

• as x-> -∞  f(x) -> ∞
   as x-> ∞  f(x) -> -∞

• as x-> -∞ f(x) -> ∞
   as x-> ∞  f(x) -> ∞

• as x->-∞ f(x) -> -∞
   as x-> ∞   f(x) -> ∞

• as x-> -∞ f(x) -> -∞
   as x-> ∞ f(x) -> -∞

Select polynomial whose zeros and degree are given.
Zeros: – 5(multiplicity of 3), 9(multiplicity of 2), – 2, 4
Degree: 7

• (x + 5)³(x - 9)²
  (x + 2)(x - 4)

• (x + 5)³(x - 9)²
  (x + 2)⁷(x - 4)⁷

• (x + 5)³(x - 9)²(x + 2)

• (x + 5)(x - 9)(x + 2)
  (x - 4)