1. 7.4 and 7.5 Solving and Zeros of Polynomials
Day 1

2. As with some quadratic equations, factoring a polynomial equation is one way to find its real roots.
Recall the Zero Product Property. You can find the roots, or solutions, of the polynomial equation P(x) = 0 by setting each factor equal to 0 and solving for x.

3. Fundamental Theorem of Algebra
Every polynomial function with degree n greater than or equal to 1 has exactly n complex zeros, including multiplicities

4. The polynomial 3x5 + 18x4 + 27x3 = 0 has two multiple roots, 0 and –3
The polynomial 3x5 + 18x4 + 27x3 = 0 has two multiple roots, 0 and –3. The root 0 is a factor three times because 3x3 = 0.
The multiplicity of root r is the number of times that x – r is a factor of P(x).
When a real root has even multiplicity, the graph of y = P(x) touches the x-axis but does not cross it.
When a real root has odd multiplicity greater than 1,
the graph “bends” as it crosses the x-axis.

5. You cannot always determine the multiplicity of a root from a graph
You cannot always determine the multiplicity of a root from a graph. It is easiest to determine multiplicity when the polynomial is in factored form.

6. Example 1A: Using Factoring to Solve Polynomial Equations
Solve the polynomial equation by factoring.
4x6 + 4x5 – 24x4 = 0
4x4(x2 + x – 6) = 0
Factor out the GCF, 4x4.
4x4(x + 3)(x – 2) = 0
Factor the quadratic.
4x4 = 0 or (x + 3) = 0 or (x – 2) = 0
Set each factor equal to 0.
x = 0, x = –3, x = 2
Solve for x.
The roots are 0, –3, and 2.

7. Example 1A Continued
Check Use a graph. The roots appear to be located at x = 0, x = –3, and x = 2. 

8. Example 1B: Using Factoring to Solve Polynomial Equations
Solve the polynomial equation by factoring.
x = 26x2
x4 – 26 x = 0
Set the equation equal to 0.
Factor the trinomial in quadratic form.
(x2 – 25)(x2 – 1) = 0
(x – 5)(x + 5)(x – 1)(x + 1)
Factor the difference of two squares.
x – 5 = 0, x + 5 = 0, x – 1 = 0, or x + 1 =0
x = 5, x = –5, x = 1 or x = –1
Solve for x.
The roots are 5, –5, 1, and –1.

9. Example 2A: Identifying Multiplicity
Identify the roots of each equation. State the multiplicity of each root.
x3 + 6x2 + 12x + 8 = 0
Since this is difficult to factor, use a graph. A calculator graph shows a bend near (–2, 0). 

10. Check to see if that is true!
The root –2 has a multiplicity of 3. Therefore x + 2 is a factor three times.
Check to see if that is true!
(x + 2)(x + 2)(x + 2)= x3 + 6x2 + 12x + 8 

11. Example 2B: Identifying Multiplicity
Identify the roots of each equation. State the multiplicity of each root.
x4 + 8x3 + 18x2 – 27 = 0
A calculator graph shows a bend near (–3, 0) and crosses at (1, 0).
The root 1 has a multiplicity of 1. The root –3 has a multiplicity of 3. Therefore (x – 1) is a factor once, and (x + 3) is a factor three times.
Check:
(x – 1)(x + 3)(x + 3)(x + 3)=x4 + 8x3 + 18x2 – 27 

12. Location Principle
If P is a polynomial function and P(x1) and P(x2) have opposite signs, then there is a real number r between x1 and x2 that is a zero of P, that is, P(r) = 0

13. Lesson Quiz
Identify the roots of each equation. State the multiplicity of each root.
0 and 2 each with multiplicity 2
1. 5x4 – 20x3 + 20x2 = 0
2. x3 – 12x2 + 48x – 64 = 0
4 with multiplicity 3
–3, 3, 1
3. x3 + 9 = x2 + 9x

14. Homework
7.4 ws