Solving Polynomials

Solving Polynomial Equations

with a factoring review

To solve a polynomial means to do all these things:

• find the solutions

• find the zeroes

• find the roots

• find the x-intercepts

Multiple Select

What are the factoring methods we have learned for factoring polynomials? (check all that apply)

• Factor out a GCF

• Difference of Squares

• Sum or Difference of Cubes

• Factor by Grouping

• Factor like a Quadratic

Let's try a problem using each of these methods.

Multiple Choice

Factor using Factoring by Grouping:

2n⁴ + 7n³ - 4n - 14

• (n - 2)(n - 2)(n - 2)(2n + 7)

• (n - 2)³(2n + 7)

• (n³ - 2)(2n + 7)

• (n + 1)(n - 1)(2n + 7)

Multiple Choice

Factor by using GCF:
8n³ + 2n²

• 2n²(4n-1)
• 2n²(4n + 1)
• n²(8n+2)
• 3n(4n+1)

Multiple Choice

Factor using Difference of Squares:  $x^2-121$  

•  $\left(x-11\right)^2$  

•  $\left(x+11\right)\left(x-11\right)$  

•  $\left(x-60.5\right)\left(x+60.5\right)$  

•  $\left(x+11\right)^2$  

Multiple Choice

Factor using Factoring like a Quadratic:

k⁴ - 10k² - 39

• (k² - 13)(k² + 3)

• (k - 13)(k + 3)

• (k² + 15)(k² - 3)

• (k + 13)(k - 3)

Multiple Choice

Factor x³ - 1000 using Difference of Cubes

• (x + 10)(x² - 10x + 100)

• (x - 10)(x² + 10x + 100)

• (x - 1)(x² + 1x + 1)

• (x - 5)(x² + 5x + 25)

Steps for solving a polynomial using factoring:

1. Factor completely (check each factor)

2. Set each factor to zero

3. Solve each factor for the variable

4. Write the solutions for the variable

Example: Solve the equation

 $x^3-x^2-30x=0$  

Factor out the GCF            $x\left(x^2-x-30\right)$  

Factor the trinomial with guess and check (find 2 factors of 30 that sum to -1)
                                           $x\left(x-6\right)\left(x+5\right)$  
Set each factor to 0 and solve:     $x=0,\ x-6=0,\ x+5=0$                   
 $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x=0,\ x=6,\ x=-5$   

Multiple Select

Solve the following polynomial. $0=x^3-3x^2-4x+12$ 

 (select all that apply)

•  $x=\pm2$  

•  $x=3$  

•  $x=1$  

•  $x=0$  

Multiple Select

Solve the following polynomial.         $x^4+16=8x^2$ 

 (select all that apply)

•  $x=0$  

•  $x=\pm2$  

•  $x=\pm4$  

•  $x=8$  

•  $x=1$  

Example: Find the zeros of the function

 $f\left(x\right)\ =\ 2x^5-450x^3$  

Replace  $f\left(x\right)$  with 0                                  $0\ =\ 2x^5-450x^3$  

Factor out the GCF                                       $0=\ 2x^3\left(x^2-225\right)$  

Factor the difference of squares                $0=2x^3\left(x-15\right)\left(x+15\right)$ 

Set each factor to 0 and solve:                  $x=0,\ \ \ x=15,\ \ \ x=-15$   

Multiple Select

Find the zeros. $g\left(x\right)=3x^3-15x^2-72x$  

(select all that apply)

•  $x=-3$  

•  $x=8$  

•  $x=-7$  

•  $x=0$  

•  $x=3$  

Multiple Select

Find the zeros:                 (select all that apply)                 

 $f\left(x\right)=5w^3-125w$  

•  $w=5$  

•  $w=-5$  

•  $w=0$  

•  $w=25$  

•  $w=5i$  

Multiple Select

Challenge problem:     Find the solutions      (select all that apply)

 $3y^3=-21y$  

•  $y=\pm7$  

•  $y=0$  

•  $y=\pm7i$  

•  $y=\pm i\sqrt{7}$  

Poll

I am feeling pretty decent about solving polynomials

• strongly agree

• agree

• meh...

• disagree

• strongly disagree