TODAY IN CALCULUS…

Warm Up: Review simplifying radicals

Learning Targets : You will use special products and factorization

techniques to factor polynomials. You will find the domains of radical expressions. You will use synthetic division to factor

polynomials of degree three or more. You will use Rational Zero Theorem to find the real

zeros of polynomials.

Independent practice

WARM UP: Simplify by removing all possible factors from the radical.

1. 2.

0.4 SPECIAL PRODUCTS AND FACTORIZATION TECHNIQUESPROPERTY

NAME DEFINITION EXAMPLE

QUADRATIC FORMULA

SPECIAL PRODUCTS

𝑎𝑥2+𝑏𝑥+𝑐=0

𝑥2+3𝑥−1=0

EXAMPLE 1: Use the Quadratic Formula to find all real zeros of each polynomial.

1. 2. 𝑎𝑏𝑐 𝑎𝑏𝑐

is imaginary…there are no real zeros!

0.4 APPLYING THE QUADRATIC FORMULA

PRACTICE: Use the Quadratic Formula to find all real zeros of each polynomial.

1. 2. 𝑎𝑏𝑐 𝑎𝑏𝑐

is imaginary…there are no real zeros!

0.4 APPLYING THE QUADRATIC FORMULA

0.4 FACTORING QUADRATICS

EXAMPLE 2: Factor then find all real zeros of the following quadratic equations.1. 2.

3. 4.

0.4 FACTORING QUADRATICS

PRACTICE: Factor then find all real zeros of the following quadratic equations.1. 2.

3. 4.

0.4 FACTORING QUADRATICS IN STANDARD FORM

EXAMPLE 1: Factor then find the real zeros of the following quadratic equation.

Multiply 1st and 3rd coefficient:

2 numbers with sum of the 2nd coefficient, but product of 30:

0.4 FACTORING QUADRATICS IN STANDARD FORM

EXAMPLE 2: Factor then find the real zeros of the following quadratic equation.

Multiply 1st and 3rd coefficient:

2 numbers with sum of the 2nd coefficient, but product of -21:

0.4 FACTORING QUADRATICS IN STANDARD FORM

EXAMPLE 3: Factor then find the real zeros of the following quadratic equation.

Multiply 1st and 3rd coefficient:

2 numbers with sum of the 2nd coefficient, but product of 24:

0.4 FACTORING QUADRATICS IN STANDARD FORM

PRACTICE: Factor then find the real zeros of the following quadratic equation.

Multiply 1st and 3rd coefficient:

2 numbers with sum of the 2nd coefficient, but product of -24:

0.4 FINDING THE DOMAIN OF A RADICAL EXPRESSION

EXAMPLE: Find the domain of

Value in square root must be positive Factor and find roots

Draw a number line and test the intervals between zeros to find where the polynomial will be positive

Define the DOMAIN using interval notation

𝟐𝟏 𝟑𝟏 .𝟓𝟎+¿ +¿−

0.4 FINDING THE DOMAIN OF A RADICAL EXPRESSION

PRACTICE: Find the domain of

Value in square root must be positive Factor and find roots

Draw a number line and test the intervals between zeros to find where the polynomial will be positive

Define the DOMAIN using interval notation

𝟏−𝟐 𝟐𝟎−𝟑+¿ +¿−

EXAMPLE 1: Factor the following quadratic equation.

STEP 1:Use the RATIONAL ZERO THEOREM

to find all possible zeros:Test all possible zeros into equation:

We know is a zero and can use SYNTHETIC DIVISION to find all other factors.

1

0.4 FACTORING POLYNOMIALS WITH DEGREE 3 OR MORE

𝑝𝑞

EXAMPLE 1: Factor the following polynomial.

STEP 2:Use SYNTHETIC DIVISION to find all other factors:

Produces the remaining polynomial:

Factor: Rewrite the polynomial

in factored form:

1

0.4 FACTORING POLYNOMIALS WITH DEGREE 3 OR MORE

𝟐1−45−2

1−210

2−42+¿ +¿ +¿

¿ (𝒙−𝟐)(𝒙 −𝟏)(𝒙−𝟏)

EXAMPLE 2: Factor the following polynomial:STEP 1:Use the RATIONAL ZERO

THEOREM:STEP 2:Use SYNTHETIC DIVISION to find all other factors:

Produces the remaining polynomial:

Factor: Rewrite the polynomial

in factored form:

1𝑥3−0 𝑥2−7𝑥−6

0.4 FACTORING POLYNOMIALS WITH DEGREE 3 OR MORE

−𝟐10−7−6

1−2−30

−2 46+¿ +¿ +¿

¿ (𝒙+𝟐)(𝒙−𝟑)(𝒙+𝟏)

1𝑝𝑞

PRACTICE: Factor the following polynomial: STEP 1:Use the RATIONAL ZERO

THEOREM:STEP 2:Use SYNTHETIC DIVISION to find all other factors:

Produces the remaining polynomial:

Factor: Rewrite the polynomial

in factored form:

0.4 FACTORING POLYNOMIALS WITH DEGREE 3 OR MORE

−𝟐2−1−13−6

2−5−30

−4106+¿ +¿ +¿

¿ (𝒙+𝟐)(𝟐𝒙+𝟏)(𝒙−𝟑)

𝑝𝑞

HOMEWORK #4:

Pg.24: 5, 9-47odd, 55-61odd, 65, 75

If finished, work on other assignments:

HW #1: Pg.7: 5-35oddHW #2: Pg.12: 3-31odd, 35-43odd, 44, 45HW #3: Pg.18: 5-51odd