RATIONAL ROOT THEOREM

​Rational Root Theorem

By Reynand Maitem

Multiple Choice

Which expression shows the factors of: n² + 16n + 63?

• 
  (n-7)(n+4)

• 
  (n-3)(n-10)

• 
  (n+7)(n-9)

• 
  (n+7)(n+9)

By Reynand Maitem

Guide questions:

What do we mean by “Factors”?

How can these help us in solving polynomial equations?

By Reynand Maitem

“Have you ever tried guessing a root of a polynomial and ended up with the wrong answer?
What if there was a method to help you narrow down the possibilities and make more accurate predictions?”

​Rational Root Theorem

By Reynand Maitem

​Objectives:
1. list of possible roots of a polynomial equation
2. Find the roots of a polynomial equation using Rational Root Theorem

By Reynand Maitem

That’s exactly what the Rational Root Theorem provides. It helps us identify all possible rational roots of a polynomial based on its coefficients, making the process of solving equations more accurate and efficient."

Why do we need to find roots of polynomials?
How can Rational Root Theorem help us solve equations more efficiently?

​The Rational Root Theorem

By Reynand Maitem

Note: The rational root theorem is used when the leading coefficient of the polynomial equation is neither zero nor one.

​The Integral Root Theorem

An integer k is a root of a polynomial equation only if k is a factor of the constant term.

Note: This theorem is applied when the leading coefficient of the polynomial equation is 1.

By Reynand Maitem

​The Integral Root Theorem An integer k is a root of a polynomial equation only if k is a factor of the constant term.

By Reynand Maitem

Guide questions:

What are the factors of the constant term?

By Reynand Maitem

By Reynand Maitem

Guide questions:

What are the factors of the constant term?
what are the factors of the leading terms?
How do we combine these to form possible rational roots?

Board work: List the possible roots of


1. p(x) = x³ − 2x² − x + 2
2. p(x) = 2x³ − 5x² + 4x − 1
3. p(x) = 3x³ − 7x² + 29x − 9

By Reynand Maitem

By Reynand Maitem

Guide questions:

How did we get these possible roots?

What do we do with these values next?

By Reynand Maitem

Testing of possibilities of roots using Factor Theorem and or Synthetic Division.

By Reynand Maitem

By Reynand Maitem

Board work: Find the rational roots of


1. p(x) = x³ − 2x² − x + 2.
2. p (x) = 2x³ − 5x² + 4x − 1.

By Reynand Maitem

By Reynand Maitem

Guide questions:

What strategy did you use to test roots?

How did you confirm your answer?

By Reynand Maitem

Finding Practical Application

By Reynand Maitem

Generalization

Multiple Choice

Which of the following are the possible roots of $$x^3+3x^2-4x-12?$$

• $$\pm1,\ \pm3,\ \pm4,\ \pm12$$

• $$\pm1,\ \pm2,\pm3,\ \pm4,\ \pm6,\pm12$$

• $$\pm1,\ \pm2,\ \pm12$$

• $$\pm3,\ \pm4,\ \pm6,\ \pm12$$

Multiple Choice

What are the roots of $$2x^3+5x^2-4x-3?$$

• $$\frac{1}{2},1,\ 3$$

• $$-\frac{1}{2},-1,\ -3$$

• $$-\frac{1}{2},1,\ -3$$

• $$-\frac{1}{2},-1,\ 3$$

Multiple Choice

What are the roots of $$2x^3-3x^2-11x+6?$$

• -2, 1/2, 3

• -6, -1/2, 2

• -1/2, 2, 3

• -1/3, 6, -3

Thank you!

By Reynand Maitem