Warm-Up

Determine the Factored Form of the following polynomial.

Synthetic Substitution and Imaginary Roots

Learning Targets

Use synthetic substitution to evaluate our polynomial at different inputs

Look at imaginary roots and determine how to write them in factored form

Learn the Sum-Product Rule to check our work and sure that our roots are correct.

Decide whether a root exists at the given value: state your reason.

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Use your calculator to evaluate the following:

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Synthetic Substitution

When polynomial p(x) is by (x-c) the remainder r will be f(c)

When f(c)=0 then x-c is a factor of the polynomial. AND

When x-c is a factor of the polynomial then f(c)=0

Factor Theorem

Factoring Sum/Product Rule

Factor the following quadratics:

A) Roots: -2,-10

B) Roots: -7,5

Sum/Product Rule

Determine the standard form of the quadratic given the roots below:

Roots: 5,6

Roots:-3,5

Roots: -2,-1

Sum/Product Rule

General Setup of the Sum and Product RuleExampleRoots: 71, 3Root 1 Root 2 Final

Sum: 71 3 74

Product:

71 3 213

𝑥2−74 𝑥+213

Quadratic with Imaginary Roots

𝑓 (𝑥 )=𝑥2+2 𝑥+4

𝑥=−2±√22−4 (1)(4)

2

𝑥=−2±√−122

𝑥=−1+√3𝑖 𝑥=−1−√3 𝑖ROOTS

Imaginary Roots

However we should always evaluate these pairs of roots to see if we are left with our original quadratic.

Simplify and Distribute

This is the quad we started with!

You try…

Write the following quadratics in their factored form. Check your answers!

a) b) c)

Activity: Polynomial Puzzle Pick a card Using that card’s instructions you have to

fill out the four section paper You will be designing your own polynomial

Don’t make it impossible and don’t make it too easy – use your best judgment!!!

After filling out the four section paper cut out each square and mark it using a symbol followed by what period you attend

Keep the four pieces together and turn them in

For Tonight:

Worksheet