5.3 Solving Poly Eqns Factor & Graph

Welcome to Chapter 5.3

We will be solving polynomial equations by...

1. Factoring

2. Graphing

.

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A few review questions first.

Multiple Select

$$x^2+6x+8=0$$

Factored form is

$$\left(x+2\right)\left(x+4\right)=0$$

Thus the solutions are which of the two below?

• x = -2

• x = 4

• x = - 4

• x=0

Multiple Select

$$x^2-25=0$$

Factored form would be

$$\left(x-5\right)\left(x+5\right)=0$$

Thus the two solutions are...

• x = 5

• x = - 5

• x = 0

• x = 3

Multiple Select

$$x^2+16=0$$

This cannot be factored but we can use square roots to solve

$$x=\pm\sqrt[]{-16}$$

We know that $$i=\sqrt[]{-1}$$

Thus the two solutions must be...

• $$x=-4i$$

• $$x=i$$

• $$x=4i$$

• $$x=16i$$

Polynomials of degree 2 will always have 2 solutions. Sometimes we call them zeros.

​Degree 3 polynomials have 3 solutions





Degree 4 polynomials have 4 solutions





Degree 5 polynomials have 5 solutions

Example one...

has two real solutions, where we MUST use the quadratic formula.
Write this example down.
The next few MC slides will use this strategy.

Multiple Select

If this is the outcome of a quadratic formula what are the solutions?

$$x=\frac{-1\pm\sqrt[]{3}}{2}$$

• $$\frac{-1+\sqrt[]{2}}{2}$$

• $$\frac{1+\sqrt[]{2}}{3}$$

• $$\frac{-1-\sqrt[]{3}}{2}$$

• $$\frac{-1+\sqrt[]{3}}{2}$$

Multiple Select

Determine the solutions by using the quadratic formula.

$$x^2-6x+4=0$$

• x = 4 +√2

• x = 4 - √2

• x = 3 +√5

• x = 3 - √5

Lets get to a polynomial of degree 4

Multiple Choice

$$\left(x-2\right)\left(x+2\right)\left(x-2\right)\left(x+2\right)=0$$

These are the factors so the solutions are choosing only one below.

• 2 & -2

• 4 & -4

• 1 & -1

• 0 & 2

Multiple Choice

How many solutions will the following polynomial have?

$$x^4+7x^2-18=0$$

• 2

• 3

• 4

• 5

Multiple Select

Factor completely to obtain the solutions

$$3x^3-6x^2-9x=0$$

Choose all that apply

• x = 0

• x = - 1

• x = 3

• x = 2

Multiple Select

Factor to solve the polynomial which means find the zeros

$$x^3-27=0$$

$$x^3-3^{3\ }=0$$

Choose all that apply

• $$\frac{-3+i\sqrt[]{3}}{2}$$

• x = 3

• $$\frac{-3+3i\sqrt[]{3}}{2}$$

• $$\frac{-3-3i\sqrt[]{3}}{2}$$

Multiple Select

Factor to solve

$$x^4+7x^2-18=0$$

• $$x=\pm3i$$

• $$x=\pm2i$$

• $$x=\pm\sqrt[]{3}$$

• $$x=\pm\sqrt[]{2}$$

Write the following equation down
You will need this for the next slide. It will take you to Desmos
1. Type this equation in and then find the x values where this crosses or touches the x-axis.
2. Write down those values as you will need them for next slide after that.

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Multiple Select

The solutions from the graph were which of the following (choose all that apply)

• -2

• 1

• 5

• 2

Let's do this again....
Write the following equation down
You will need this for the next slide. It will take you to Desmos
1. Type this equation in and then find the x values where this crosses or touches the x-axis.
2. Write down those values as you will need them for next slide after that.

Home

Open in new tab

Copy link

Multiple Select

The solutions from the graph were which of the following (choose all that apply)

• -2

• -1

• 5

• 0

Multiple Select

Solve by factoring

$$x^4-64=0$$

• $$x=\pm2i\sqrt[]{2}$$

• $$x=\pm2\sqrt[]{2}$$

• $$x=\pm\sqrt[]{2}$$

• $$x=\pm2i$$

1. Solving a polynomial means to find the real number values where the graph crosses or touches the x-axis

2. ​We call these zeros, or solutions.

3. We may have to factor and factor more than once

4. We might need the quadratic formula even if we have factored once.

5. We can have some real or some imaginary or both types.

6. If we are using a graph to solve, the graph will only show the real zeros.

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Summary