Solving Polynomials

Presentation

•

Mathematics

•

9th - 12th Grade

•

Practice Problem

•

Medium

•

CCSS

HSA.SSE.A.2, HSA.APR.C.4, HSA.APR.B.3

Standards-aligned

Bonnie Keating

Used 157+ times

FREE Resource

6 Slides • 12 Questions

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

Solving Polynomial Equations

with a factoring review

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