GEM T2 C3S4-5 RRT Solving Polynomial Equations

Multiple Choice

Which of the following best describes the significance of the Rational Root Theorem in algebra?

• It helps identify all possible rational roots of a polynomial equation.

• It guarantees that every polynomial has a rational root.

• It is used to factor quadratic equations only.

• It determines the degree of a polynomial.

Open Ended

Why is it important to determine the possible roots of a polynomial equation before attempting to solve it?

Multiple Choice

According to the Fundamental Theorem of Algebra, how many roots does a polynomial equation of degree n have, counting multiplicity?

• n

• n+1

• n-1

• 2n

Multiple Select

Which of the following are valid methods for solving quadratic equations?

• Factoring

• Completing the square

• Using the quadratic formula

• Synthetic division

Open Ended

Explain the difference between solving incomplete and complete quadratic equations. Provide an example for each type.

Multiple Choice

What are the roots of the equation x^3 + 6x^2 - 4x - 24 = 0?

• 1, -1, 2

• 2, -2, -6

• 3, -3, 6

• 4, -4, 5

Multiple Choice

If a rational number L/F is a root of a polynomial equation with integer coefficients, what must be true about L and F?

• L is a factor of the constant term and F is a factor of the leading coefficient

• L is a factor of the leading coefficient and F is a factor of the constant term

• L and F must both be prime numbers

• L and F must both be negative

Fill in the Blanks

Fill in the blank: In the Rational Root Theorem, possible rational roots are given by the factors of the

term over the factors of the first term.

Multiple Choice

What are the possible rational roots of the equation x^3 + 6x^2 + 11x + 6 = 0 according to the Rational Root Theorem?

• 1, -1, 2, -2, 3, -3, 6, -6

• 1, 2, 3, 6

• -1, -2, -3, -6

• 0, 1, -1, 2

Open Ended

According to the Fundamental Theorem of Algebra, how many roots does the equation x^3 + 6x^2 + 11x + 6 = 0 have, and what methods can be used to find them?

Fill in the Blanks

The expression obtained after dividing x^3 + 6x^2 + 11x + 6 by (x + 1) is x^2 +

x + 6.

Open Ended

Explain how synthetic division is used to find the roots of a cubic polynomial, using x^3 + 6x^2 + 11x + 6 as an example.

Multiple Select

Which of the following are roots of the equation x^3 + 6x^2 + 11x + 6 = 0?

• -1

• -2

• -3

• 1

Multiple Choice

What are the possible rational roots of the equation x^3 - 3x^2 - 33x + 35 = 0 as determined by the Rational Root Theorem?

• 1, -1, 5, -5, 7, -7, 35, -35

• 1, -1, 2, -2, 3, -3, 6, -6

• 1, 5, 7, 35

• -1, -5, -7, -35

Multiple Choice

Which of the following are possible rational roots for the equation x^3 - 3x^2 - 33x + 35 = 0 according to the Rational Root Theorem?

• 1, -1, 5, -5, 7, -7, 35, -35

• 2, -2, 10, -10, 14, -14, 70, -70

• 1, 2, 3, 4, 5, 6, 7, 8

• 0, 1, -1, 2, -2, 3, -3, 4

Open Ended

Describe the process of using synthetic division to test possible roots for a cubic polynomial equation.

Multiple Choice

Which of the following are the roots of the equation x^3 - 3x^2 - 33x + 35 = 0?

• 1, -5, 7

• 1, -1, 5

• -1, 5, 7

• -5, 7, 35

Fill in the Blanks

After using synthetic division on x^3 - 3x^2 - 33x + 35, the resulting quadratic equation is x^2 -

x - 35 = 0.

Multiple Choice

What are the possible rational roots for the equation x^3 - 11x^2 - x + 11 = 0 according to the Rational Root Theorem?

• 1, -1, 11, -11

• 1, -1, 5, -5

• 1, 2, 11, 22

• 1, -1, 7, -7

Multiple Choice

Which of the following is NOT a possible rational root of the equation 2x^4 + x^3 - 8x^2 - x + 6 = 0 based on the Rational Root Theorem?

• ±1

• ±2

• ±3

• ±4

Multiple Select

Which of the following are roots of the polynomial 2x^4 + x^3 - 8x^2 - x + 6 = 0?

• -2

• 1

• -1

• 2

Open Ended

Explain the process of using synthetic division to find the roots of a polynomial equation, using the example 2x^4 + x^3 - 8x^2 - x + 6 = 0.

Fill in the Blanks

Fill in the blank: The remainder is

when -1 is substituted into the polynomial 2x^3 - 3x^2 - 3x + 2.

Open Ended

For how many positive integers n is n^3 - 8n^2 + 20n - 13 a prime number?

Multiple Select

Which of the following methods can be used to solve for the actual roots of a polynomial as mentioned in the synthesis slide?

• Synthetic division

• Quadratic formula

• Completing the square

• Factoring

Open Ended

Reflecting on today’s lesson, what questions do you still have about the Rational Root Theorem or solving polynomial equations?

Multiple Choice

How can the Rational Root Theorem help you solve polynomial equations?

• It helps identify possible rational roots to test for solutions.

• It provides the exact roots of any polynomial equation.

• It only works for quadratic equations.

• It is used to factor trinomials only.