Solving Polynomial Equations and Factoring

​FACTORING & SOLVING POLYNOMIAL EQUATIONS

​Dr. Tom Giles

​Over the next few slides, we'll review how to factor a quadratic equation from standard form to factored form. In addition, you will learn how to take vertex form and convert it to standard form. Furthermore, you will review how to multiply polynomials together. Lastly, you will review how to find the solutions to the quadratic equation. Let's review...

​Standard Form

​Factored Form

​Vertex Form

​Notice, all three function create the same parabola.

Red=Factored= Reveals zeros. {-3, 5}

Blue=Standard= Reveals y-intercept (0, -15)

Green=Vertex= Reveals the vertex. (1, -16)

​Quadratic Function Factoring and Solving Examples:

​Solutions:

​Standard Form

Factored Form

​The solution set of x=-9 and x=9 is written as:

​Solutions, x-intercepts, zeros and roots all mean the same thing. They are where the parabola crosses the x-axis.

​Example Problem: Quadratics Factoring and Solving

​What are the solutions to the following quadratic equations?

Match

Match the following quadratic expressions from their standard form to their factored form.

• $$x^2+10x+25$$

• $$x^2-16$$

• $$x^2+8x+16$$

• $$x^2-25$$

• $$x^2+8x+12$$

• $$\left(x+5\right)\left(x+5\right)$$

• $$\left(x+4\right)\left(x-4\right)$$

• $$\left(x+4\right)^2$$

• $$\left(x-5\right)\left(x+5\right)$$

• $$\left(x+2\right)\left(x+6\right)$$

Match

Match the following factored quadratic expressions to their standard form.

• $$\left(2x-1\right)\left(x+2\right)$$

• $$\left(3x-1\right)\left(x-4\right)$$

• $$\left(3x+1\right)\left(x-4\right)$$

• $$\left(2x+1\right)\left(x-2\right)$$

• $$\left(3x+1\right)\left(x-2\right)$$

• $$2x^2+3x-2$$

• $$3x^2-13x+4$$

• $$3x^2-11x-4$$

• $$2x^2-3x-2$$

• $$3x^2-5x-2$$

Factoring Trinomials using the AC Method

Drag and Drop

Which expression is equivalent to $$\left(z+5\right)\left(z^2-4z+6\right)$$ ?

​

Placeholder

Drag these tiles and drop them in the correct blank above

Multiple Choice

To factor 3x² + 14x + 8, find the 2 clue numbers first. The clue numbers (ac & b) are:

• a⋅c =11 and b=14

• a⋅c =14 and b=3

• a⋅c =42 and b=8

• a⋅c =24 and b=14

​Quadratic Functions: Changing from Vertex Form to Standard Form.

​Vertex Form

​Standard Form

Multiple Choice

To factor the trinomial 6x² + 11x +3, start by multiplying "a" times "c" and identifying "b".

• true

• false

​How to convert vertex form to standard form.

​Final Answer

Multiple Choice

To factor the trinomial  $$6x^2+11x+3$$   ...

• you need to find two numbers that multiply to 3 and add to 66

• you need to find two numbers that multiply to 18 and subtract to 11

•  you need to find two numbers that multiply to 18 and add to 11

• you need to find two numbers that multiply to 18 and add to 3

Dropdown

A quadratic function is defined as:

$$f\left(x\right)=\left(x-4\right)^2+3$$

What is the equation of the parabola in standard form?

​

Select answer

Multiple Choice

To factor the trinomial  $$6x^2+11x+3$$  , find 2 numbers that multiply to 18 and add to 11. Those two numbers are:

• 9 and -2

• 9 and 2

• -9 and -2

• -9 and 2

Math Response

A quadratic is defined as $$y=\left(x+3\right)^2-6$$

What is the equation for this function in standard form?

Your answer needs to be in the form of:

$$y=ax^2+bx+c$$

Enter your answer in the space provided.

Type answer here

Correct Answer: y=x^2+6x+3

Deg^°

Rad

Multiple Choice

Which expression is equivalent to $$16w^2+24w+9$$ ?

• $$\left(4w+3\right)^2$$

• $$\left(4w-3\right)^2$$

• $$\left(8w+3\right)^2$$

• $$\left(8w-3\right)^2$$

Multiple Choice

Factor:
 b² + 8b + 7

• (b + 4)(b + 3)
• (b + 6)(b + 1)
• (b + 7)(b +1)
• prime

Math Response

What is the product of the expression:

$$\left(2x+5\right)\left(x-4\right)$$ ?

Your answer should be in the form of:

$$ax^2+bx+c$$

Type answer here

Correct Answer: 2x^2-3x-20

Deg^°

Rad

​Example problem: Factoring Quadratics with a leading coefficient greater than 1.

​Array Method

​First

​Outside

Inside

Last

​Factors