01
02
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NCE
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04
03
participate.
ONE-MINUTE
PERFORMANCE TASK
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accomplish your
Explanation of problems in
OMP for your PLUS point.
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our Lesson
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Task)
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on your OWN
artwork/design using formulas in our
lessons.
Module 8
Polynomial
Equations
Module 8: Polynomial Equations
Illustrate Polynomial Equations
Find the number of roots of Polynomial
Equations
Create Polynomial Equations
Recall
Determine the number of positive & negative roots of 2x3
– 5x2 – 14x + 8, then find all the roots/zeros.
P(x) = 2x
+ 3 – 5x–2 – 14x –+ 8
+
2nd
1st
3
2
3
2
5(-x)
– 14(-x)
– –– 5x
P(-x) = –2(-x)
2x
++ 8
–+ 14x
1st
𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑟𝑜𝑜𝑡𝑠 = 2 𝑜𝑟 0
𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑟𝑜𝑜𝑡𝑠 = 1
+ 8+
Recall
𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒
=
2
𝑜𝑟
0
𝑟𝑜𝑜𝑡𝑠
𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒
=
1
𝑟𝑜𝑜𝑡𝑠
𝑃 = 1, 2, 4 , 8
𝑄 = ±1, ±2
𝑃
1
= 1, 2, 4, 8, 
𝑄
2
2x3 – 5x2 – 14x + 8
-2
2
𝟏
𝟐
–5
-4
–14 8
18 –8
2
–9
1
4
–4
4
2
–8
8
0
0
2
-2,
0
1
__
,24
Recall
A polynomial expression P(x) is an expression of the
form
n
n-1
n-2
Anx + An-1x + An-2x + …+ A1x + A0
where the non-negative integer n is called the degree of
the polynomial and the coefficients A0, A1, …, An are real
numbers and An ≠ 0.
Polynomial Equations
A polynomial equation is an equation of the form
n
anx
+
n-1
an-1x +
n-2
an-2x
+ …+ a1x + a0 = 0
where the non-negative integer n is called the degree of
the polynomial and the coefficients a0, a1, …, an are real
numbers and an ≠ 0.
Polynomial Expression
n
n-1
n-2
anx + an-1x + an-2x + …+ a1x + a0
Polynomial Equation
n
n-1
n-2
anx + an-1x + an-2x + …+ a1x + a0 = 0
Polynomial Equation
5x – 7 = 0
Linear equation
(x + 2) (3x – 4) = 0 Quadratic equation
5x3 – 3x = 0
Cubic equation
4x4 – 64 = 0
Quartic equation
(3x – 1) x = 0
(2x +
2
1)
–4=0
Module 8: Polynomial Equations – Illustrate Polynomial Equations
Tell whether if the following is a polynomial equation (PE) or NOT.
1. 3x – 4 = 0
2. y = 4x4 – 64
2
3. x(3x – x + 4) = 0
2
4. f(x) = (2x + 1) – 4
PE
NOT
PE
NOT
Module 8: Polynomial Equations – Illustrate Polynomial Equations
Tell whether if the following is a polynomial equation (PE) or NOT.
-3
5. 3x – 4 = 0
6. 0 = 4x4 – 64
NOT
PE
2
7. – 2x(x – x + 7) = 0
PE
2
8. P(x) = (x – 1) – 1
NOT
Recall
ZERO-PRODUCT PROPERTY
If a  b = 0, then a = 0 or b = 0
(2x)(3y) = 0
2x = 0
x(4x – 1) = 0
x=0
or
3y = 0
or 4x – 1 = 0
Polynomial Equation
Determine the real root(s) of each equation.
1. (x + 1) (x – 3) = 0
or x – 3 = 0
x+1=0
x
=
3
x
=
–1
-1
3
{-1, 3}
2. x2 + x – 2 = 0
{-2, 1}
(x + 2) (x – 1) = 0
or x – 1 = 0
x+2=0
x = 11
x = –2–2
Polynomial Equation
Determine the real root(s) of each equation.
3. 𝑥2 − 6𝑥 + 9 = 0
(x − 3)2 = 0
x−3=0
x = 33
(x − 3) (x − 3) = 0
x−3=0
x = 33
or
x−3=0
x = 33
{3 of multiplicity 2}
Polynomial Equation
Determine the real root(s) of each equation.
4. x2( x – 9)(2x + 1) = 0
or 2x + 1 = 0
or x – 9 = 0
x2 = 0
x = 00
x = 99
2x = -1
1
_
_
x=
2
1
__
{
,0
of multiplicity 2
2
,9 }
Polynomial Equation
Determine the real root(s)
of each equation.
real root(s)
5. (x + 4) (𝑥2 − 𝑥 + 3) = 0
x+4=0
x = −4
−4
or
{−4}
𝑥2 − 𝑥 + 3 = 0
Unreal roots
Polynomial Equation
Determine the real root(s) of each equation.
6. (3𝑥 + 1) 2(𝑥 + 7)(𝑥 − 2)4 = 0
3𝑥 + 1 = 0
2(𝑥 + 7) = 0
𝑥
+
7
=
0
3x = −1
𝑥 = −7
1
−7
_
_
x=
3
{
(𝑥 − = 0
𝑥−2=0
1
_
_
−7,
, 2 of multiplicity 4
3
4
2)
𝑥 = 22
}
Module 8
Polynomial
Equations
Polynomial Equations
A polynomial equation is an equation of the form
n
anx
+
n-1
an-1x +
n-2
an-2x
+ …+ a1x + a0 = 0
where the non-negative integer n is called the degree of
the polynomial and the coefficients a0, a1, …, an are real
numbers and an ≠ 0.
Recall
ZERO-PRODUCT PROPERTY
If a  b = 0, then a = 0 or b = 0
(2x)(3y) = 0
2x = 0
x(4x – 1) = 0
x=0
or
3y = 0
or 4x – 1 = 0
Polynomial Equation-Practice
Determine the real root(s) of each equation.
1. (𝑥 + 3) (𝑥 − 2) (𝑥 + 1) = 0
{ -3, -1, 2 }
2. (2𝑥 + 3) (3𝑥 − 2) (𝑥 –
3
1)
_ _3 , 2_ ,
2 3
}
3. 12𝑥3 + 6x2 – 18x = 0
=0
1 of multiplicity 3
_{ _3 , 0, 1 }
2
}
Polynomial Equation
n
Anx
n-1
n-2
+ An-1x + An-2x
+ …+ A1x + A0 = 0
3x – 1 = 0
2
x +x–6=0
3
-5x
–
4
x
2
2x
–
–x+2=0
2
13x
+ 36 = 0
Polynomial Equation
Write the polynomial equation given its roots/zeros.
Zero-Product Property
1. x = –2 ,
x+2=0
x=1
x–1=0
(x + 2) (x – 1) = 0
+ 2x
2x
– 2 =–02 = 0
2+x–2=0
2
x
x +x–2=0
2
x ––xx+
If ab
a ==00,orthen
b=
0,
0.
a =then
0 orab
b == 0.
Polynomial Equation
Write the polynomial equation given its roots/zeros.
Zero-Product Property
2. roots: 2 and –3
x = –3
x=2
x–2=0
x+3=0
(x – 2) (x + 3) = 0
2
3x– 2x– –2x6 =–06 = 0
x ++3x
2
2
x +x x+ –x –66==00
If ab
a ==00,orthen
b=
a =then
0,
0 orab
b == 0.
0.
Polynomial Equation
Write the polynomial equation given its roots/zeros.
3. {3 of multiplicity 2}
x=3
x=3
x–3=0
x–3=0
(x – 3) (x – 3) = 0
x2 – 3x – 3x + 9 = 0
2
x
2 – 6x + 9 = 0
x
– 6x + 9 = 0
Converse
Zero-Product Property
If a = 0 or b =
0, then ab = 0.
Polynomial Equation
Write the polynomial equation given its roots/zeros.
,x=1
4. x = –1
3
x
–
5. x = –3
4
x
–
2
2x
,x=2
–x+2=0
, x = –2
2
13x
,x=2
+ 36 = 0
,x=3
Polynomial Equation - Practice
Write the polynomial equation given its roots/zeros.
1. x = –2
,x=4
2. x = –1
3. x = 1
,x=5
, x = -2
, x = 1/2
4. x = 1
, x = –1
,x=0
5. x = 1
, x = -1
,x=5 3
, x = –5
, x = –4
Polynomial Equation - Practice
Write the polynomial equation given its roots/zeros.
1. x = –2
,x=4
2
𝑥
2. x = –1
− 2𝑥 – 8 = 0
,x=5
2
𝑥
− 4𝑥 – 5 = 0
Polynomial Equation - Practice
Write the polynomial equation given its roots/zeros.
, x = -2
3. x = 1
3
2𝑥
4. x = 1
5
𝑥
+
+
2
𝑥
− 5𝑥 + 2 = 0
, x = –1
4
9𝑥
+
, x = 1/2
3
19𝑥
,x=0
−
2
9𝑥
, x = –5
− 20𝑥 = 0
, x = –4
Polynomial Equation - Practice
Write the polynomial equation given its roots/zeros.
, x = -1 , x = 5  3
5. x = 1
4
𝑥
−
3
10𝑥
+
2
12𝑥
+ 10𝑥 − 13 = 0
Polynomial Equation - Practice
Solve word problems with polynomial equations.
1. The sum of a number and its square is 72. Find the
number.
2. The area of a triangle is 44𝑚2. Find the lengths of
the legs if one of the legs is 3m longer than the other
leg.
Solve word problems with polynomial equations.
its
1. The sumsum
of a numberaand number
its square is 72. Find the
number. is 72
square
Representation:
x = number
Equation:
x + x2 = 72
Solution:
2
x
+ x = 72
x2 + x – 72 = 0
( x –) 8 ( x +) 9 = 0
x–8=0
x+9=0
x = –9
x=8
1. The sum of a number and its square is 72. Find the
number.
Representation:
x = number
Equation:
x + x2 = 72
Checking:
Solution:
x2 + x = 72
x2 + x – 72 = 0
( x –) 8 ( x +) 9 = 0
x–8=0
x+9=0
x = –9
x=8
?
–98 + 81
64 = 72
72 = 72 The number is 8 or -9.
Solve word problems with polynomial equations.
2
44𝑚 .
2. The area of a right triangle is
Find the
lengths of the legs if one of the legs is 3m longer than
the other leg.
2. The area of a right triangle is 44𝑚2. Find the lengths
of the legs if one of the legs is 3m longer than the other
leg.
Solution:
Representation:
1
_
x = one of the legs
44 = (x) (x + 3)
2
x + 3 = other leg
1
_
Equation:
44 = (x2 + 3x)
2
1 bh
_
A =
2
88 = x2 + 3x
1
_
44 = (x) (x + 3)
2 + 3x – 88
0
x
=
2
triangle
2. The area of a right triangle is 44𝑚2. Find the lengths
of the legs if one of the legs is 3m longer than the other
leg.
Checking:
2
0 = x + 3x – 88
? _1
44 = (8) (8 + 3)
2
0 = ( x –) 8 ( x +)11
The length of the
? _1
44 are
legs
and 11
(8) (11)
x + 11 = 0
x–8=0
= 28𝑚
x = –11
x=8
? _1𝑚.
44 = (88)
Not possible
2
other leg
negative length
44 = 44
8 + 3 = 11