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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
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