Pre-Calculus 12 Notes

By: Gen L.

Lesson 4: Rational Zero Theorem

Rational Zeros Theorem

Sometimes, zeros could be rational if the lead coefficient isn't 1.

if .

Find Rational Roots

1. Find all possible Rational roots of .

• This gives us .

2. Factor

We get:

Factoring Practice:

• We have two different paths: Factor by Inspection ("The Shortcut") and Decomposition.

• • Factor by inspection:

• • Decomposition:

How To Factor

You now have another possible way added to your toolbelt:

• Product / Sum (The First Path)
• Diff. of Squares (Binomials only)
• PST ()
• Decomposition ()
• Inspection ("The Shortcut")

Recognising which method to use makes factoring easier.

Example:

Factor

1. Determine all possible Rational Roots

2. Synthetic divide to simplify

Example, cont.

We have:

3. Simplify, if possible.
   • Using factoring by inspection on the trinomial, we get

Result:

Determining Coefficients

Determine , s.t. is a factor of .

Applications Example 1

The Volume of a prism is given by . If the height is , what polynomials represent the possible length and width.

• Common Assumption: .

Applications Example 1, cont.

We now know that

• Finally, factor to get our values.

Applications Example 2

A boardwalk that is feet wide is built around a rectangular pond. The Pond is . The combined surface area is . What is the width of the boardwalk.

• Try drawing a picture to help you.

Applications Example 2 Answer

• What we Know:
• Solve:

Applications Example 2, cont.

• We have:
• Continuing solving:

• This gets us:
• And finally, our result: .

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