Lesson 9.4 Solve Polynomial Equations in Factored Form Essential Question: How do you solve polynomial equations in factored form?

Before we start… What are the x-intercepts of the graph? Consider the equation 𝑥−2 𝑥−6 =0. Substitute the intercepts into the equation. How are they related to the solutions of the equation?

What is the Zero Product Property? If two numbers multiplied equals 0, then ONE of the numbers has to be 0. If 𝑎𝑏=0, then 𝑎=0 or 𝑏=0.

Why do we need the zero product property? It allows us to solve equations when one side is equal to 0. The solutions of the equations are called roots.

How do I solve an equation in factored form? Set the equation equal to 0. Set each factor equal to 0. Solve each factor.

Solve 𝑥−4 𝑥+2 =0

Solve 𝑥−5 𝑥−1 =0

Solve 𝑥+3 𝑥−5 =0

Solve 𝑦+9 𝑦−1 =0

Solve 3𝑥+1 𝑥+6 =0

What is factoring? Factoring is undoing the multiplication of polynomials. You will use factoring to solve polynomial equations.

What are the different ways to factor? Factoring Trinomials 𝒙 𝟐 +𝒃𝒙+𝒄 𝒂 𝒙 𝟐 +𝒃𝒙+𝒄 Difference of Two Squares Perfect Square Trinomial Greatest Common Factor

How do you factor a greatest common factor? Find the greatest common factor of the terms Number, variable, number and variable Pull the GCF out front Divide each term by the GCF and write what remains

Find the GCF 12𝑥+42𝑦

Find the GCF 4 𝑥 4 +24 𝑥 3

Find the GCF 14 𝑦 2 +21𝑦

How do you solve a polynomial equation by factoring? Set the equation equal to 0. Factor the polynomial. Set each factor equal to 0. Solve each factor.

Solve 2 𝑥 2 +8𝑥=0

Solve 6 𝑛 2 =15𝑛

Solve 3 𝑠 2 −9𝑠=0

Solve 4 𝑥 2 =2𝑥

Solve 𝑞 2 +16𝑞=0

Solve 12 ℎ 2 =36ℎ

Solve 6 𝑥 2 +12𝑥=0

Solve 9 𝑦 2 =21𝑦

What is vertical motion? The path that a projectile, or object, takes. The height of a projectile can be described by the vertical motion model.

Vertical Motion Model The height h (in feet) of a projectile can be modeled by ℎ=−16 𝑡 2 +𝑣𝑡+𝑠 where t is the time (in seconds) the object has been in the air, v is the initial vertical velocity (in feet per second), and s is the initial height (in feet).

Vertical Motion Model The height h (in meters) of a projectile can be modeled by ℎ=−4.9 𝑡 2 +𝑣𝑡+𝑠 where t is the time (in seconds) the object has been in the air, v is the initial vertical velocity (in meters per second), and s is the initial height (in meters).

Armadillo A startled armadillo jumps straight into the air with an initial vertical velocity of 14 feet per second. After how many seconds does it land on the ground?

Dolphin A dolphin jumped out of the water with an initial velocity of 32 feet per second. After how many seconds did the dolphin enter the water?

Jump Rope A child jumping rope leaves the ground at an initial vertical velocity of 8 feet per second. After how many seconds does the child land on the ground?

Kangaroo A kangaroo jumps from the ground with an initial vertical velocity of 18 feet per second. How long is the kangaroo in the air for?

Kickball A kickball is kicked upward with an initial vertical velocity of 3.2 meters per second. The height of the ball is given by ℎ=−4.9 𝑡 2 +3.2𝑡 where h is measured in meters and the time t is measured in seconds. After how many seconds does the ball land?

Arrow An arrow that is fired straight up will fly to a greater height than an arrow fired at an angle. Robin fires an arrow from the ground at an unknown angle with an initial velocity of 75 meters per second. How long will it take for the arrow to hit the ground?

Rabbit High Jump A rabbit in a high jump competition leaves the ground leaves the ground with an initial vertical velocity of 4.9 meters per second. How long is the rabbit in the air?

How do you solve polynomial equations in factored form?

Ticket Out the Door Solve 𝑥+2 𝑥−7 =0