chapter 1: polynomial
TRANSCRIPT
CHAPTER OVERVIEW
POLYNOMIAL
1.1 Basic concept (W1)
1.2 Algebra operation for polynomial (W1)
1.3 Remainder theorem and factor theorem (W2)
1.4 Synthetic divider (W2)
1.5 Solving Quadratic equation (W3)
1.6 Solving Inequality (linear, quadratic, absolute)
(W3)
1.1 Basic Concept
Monomials are terms that contain variables with only
whole numbers as exponent
Polynomial is made up from a monomial or finite sum
of monomials.
-3y25x 14ab3 Term
Literal
factor
Numerical
coefficiant
1.1 Basic Concept (W1)
A polynomial is an algebraic expression in which the exponents of the variables are nonnegative integers and there are no variables in the denominator
• An expression that can be written in the form
• a + bx + cx2 + dx3 +ex4 + ….
• Things with Surds (e.g. x + 4x +1 ) and reciprocals (e.g. 1/x + x) are not polynomials
• The degree is the highest index
• e.g 4x5 + 13x3 + 27x is of degree 5
Set-up a multiplication table3x2 -5x -3
2x4
+4x2
6x6 -10x5 -6x4
+12x4 -20x3 -12x2
Gather like terms = 6x6 - 10x5 + 6x4 - 20x3 -12x2
They can be added: 2x2 - 5x - 3 + 2x4 + 4x2 + 2x = 2x4 + 6x2 -3x -3
(2x4 + 4x2)(3x2 - 5x - 3)Or multiplied:
6x6 -10x5 -6x4 +12x4 -20x3 -12x2
Polynomials can be combined to give
new polynomials
Function Notation
• An polynomials function can be written as
• f(x) = a + bx + cx2 + dx3 +ex4 + ….
• f(x) means ‘function of x’
• instead of y = ….
• e.g f(x) = 4x5 + 13x3 + 27x
• f(3) means ….
– “the value of the function when x=3”
– e.g. for f(x) = 4x5 + 13x3 + 27x
– f(3) = 4 x 35 + 13 x 33 + 27 x 3
– f(3) = 972 + 351 + 81 = 1404
x
f(x)
Example
Identify which expressions are polynomials. If an expression is
not a polynomial, explain why.
(a)
(b)
(c)
(d)
235 2 ++ xx
xx
35 −
9
232
1 −+− xx
Some polynomials have special names depending on
the number of terms contained in the polynomial.
A monomial is a polynomial containing one term. A
term may have more than one factor.
A binomial is a polynomial containing two terms.
A trinomial is a polynomial containing three terms.
4
3,7,5,2,3
22 a
xyabx−
4,52,3 2 yxxxx +−+
57
2,43, 2 −++−++
axxxcba
14 +x
83 3 −x
1425 2 −+ xx
2
Polynomials in one variable
The degree of a polynomial in one variable is the largest exponent of that variable.
A constant has no variable. It is a 0 degreepolynomial.
This is a 1st degree polynomial. 1st degree polynomials are linear.
This is a 2nd degree polynomial. 2nd degree polynomials are quadratic.
This is a 3rd degree polynomial. 3rd degree polynomials are cubic.
Classify the polynomials by degree and number of terms.
Polynomial
a.
b.
c.
d.
5
42 −x
xx +23
14 23 +− xx
Degree
Classify by
degreeClassify by
number of terms
Zero Constant Monomial
First Linear Binomial
Second Quadratic Binomial
Third Cubic Trinomial
Example
• Try this:
• Identify the degree of each term in the polynomial
Identify the Degree of Terms
Exponents of variable factors must be integers greater than 0.
For a term that is constant, the degree is 0.
For a term that has only one variable factor, the degree is the same as the exponent of the variable.
For a term that has more than one variable factor, the degree is the sum of the exponents of the variables.
3325 23 +−+ xxx
• A linear term has degree 1
• A quadratic term has
degree 2
• A cubic term has degree 3
Identify the Degree of Polynomials
Identify the degree of each term of the polynomial.
Compare the degrees of each term of the
polynomial and select the greatest degree as the
degree of the polynomial.
• Try this:
• Identify the degree of the polynomial
• (a) has a degree of 2
• (b) has a degree of 3 and is a cubic poly
• (c) has a degree of 0 and is a constant
• (d) has a degree of 1 and is a linear polynomial
• (e)
2
1
7
543
125
23
2
−
−+−
−+
x
xxx
xx
𝑥2 +2
𝑦
1.2 Basic Operations with Polynomials
(1) Add and Subtract Polynomials
Combine the coefficients of the like terms using the rules for adding or subtracting signed numbers
(2) Multiply Polynomials
Multiply the coefficients using the rules for signed numbers.
Multiply the letter factors using the laws of exponents for factors with like base
Distribute if multiplying a polynomial by a monomial
(3) Divide Polynomials
Divide the coefficients using the rules for signed numbers.
Divide the variable factors using the laws of exponents for factors with like base
24
4
x
x−
( )22 332 yyyy +−+
( )xxx 42 2 −
Take a break
• What does simplify mean?
• - instruction to ‘simplify the expression’ are vague but
often used.
• To simplify an expression means to write the expression
using fewer terms or reduced or with lower coefficients
or exponents.
• Examine the expression and see which laws or operations
allow you to rewrite the expression in a simpler form.
Exercises
Simplify the algebraic expressions by combining
like terms.
(a) (e)
(b) (f)
(c) (g)
(d) mnm
aaaa
xx
xx
−+
−++
−
+
53
634
4
25
2323
55
33
( )
( ) ( )2524453
532
5
2323
22
32
+−−−+−−
+−+
+−+
mmmmmm
yyyy
yyyy
Exercises (multiply polynomials)
( )( )
( )( )
( )( )
( )( )
( )( )yaya
xx
aa
yy
xx
−+
++
−−
−
22
24
3
26
34
32
2
( )
( )
( )
( )( )653232
23
52
2
22
2
2
2
+−−+
−
−
+
xxxx
ba
x
x
Exercise (Divide polynomials)
Express the answers with positive exponents
x
xe
x
xd
x
xc
x
xb
y
ya
4
3
2
4
2
3
5
2.
12
3.
15
5.
4
4.
2
6.
−
−
−
2
23
3
23
234
2
26.
3.
3
1291518.
x
xxh
x
xxg
a
aaaaf
+
−
−−−
Divide polynomials (cont..)
There are two procedures for dividing polynomials:
1. Long division
2. Synthetic division
These procedures are especially valuable in
factoring and finding zeros of polynomial functions.
Break-even points in financial applications.
Intervals where polynomial functions are positive,
negative.
Divide P(x) = x3 – 7x – 6 by x – 4
P(x) = x3 – 7x – 6 is a dividend; Q(x) = x – 4 is a divisor
1. The divisor goes on the outside of the
box. The dividend goes on the inside of the
box.
2. Divide 1st term of dividend by first term
of divisor to get first term of the quotient.
Line up the first term of the quotient with
the term of the dividend that has the same
degree.
3. Take the term found in step 2 and
multiply it times the divisor.
4. Subtract this from the line above.
5. Repeat until done (until the degree of
the "new" dividend is less than the degree
of the divisor).
x3 –0x2 - 7x – 6x – 4
x3 –0x2 - 7x – 6x – 4
x2
x3 –0x2 - 7x – 6x – 4
x2
x3 –4x2
4x2 - 7x
1. Long division
23
Divide P(x) = x3 – 7x – 6 by x – 4
1. Long division (cont..)
P(x) = D(x)Q(x) + R
dividend =
divisor x quotient + remainder
P(x) = D(x)Q(x) + R
x3 – 7x – 6 =
(x - 4) (x2 + 4x + 9) + 30
x3 –0x2 - 7x – 6x – 4
x2 + 4x + 9
x3 –4x2
4x2 - 7x
4x2 – 16x
9x – 6
9x –36
30
Take a break
• Divide using long division and use the result to
factor the polynomial completely.
( )( )4
12175 2
−
−−
x
xx
Synthetic Division is a special case of dividing by a linear
factor binomial (x - k) or (jx - k) if x is variable, and j = const,
k = const.
Divider is a constant , which turns a binomial divisor into zero:
for a divisor (x – k) use the divider k;
for a divisor (jx – k) use the divider k/j.
Divide x4 - 3x + 5 by (x - 4),
(x- 4) is a divisor, then divider is 4.
2. Synthetic Division
2. Synthetic Division (cont..)
Divide x4 - 3x + 5 by (x - 4)
1. Write dividend in descending powers and insert 0's for any missing
terms x4 + 0x3 + 0x2 - 3x + 5
2. Find the number which turns the divisor into 0: x- 4 = 0, if x = 4
3. Write the coefficients of the dividend to the right:
4| 1 0 0 -3 5
4. Bring down the leading coefficient to the bottom row
4| 1 0 0 -3 5
1
5. Multiply c by the value just written on the bottom row
4| 1 0 0 -3 5
4 16 64 244
1 4 16 61 249
Note; Vertical
pattern : add
terms Diagonal
pattern :
Multiply by k
4| 1 0 0 -3 5
4 16 64 244
1 4 16 61 249
The numbers in the last row make up your coefficients of the
quotient as well as the remainder:
quotient = x3 + 4x2 + 16x + 61 remainder = 249
x4 - 3x + 5 = (x - 4)*(x3 + 4x2 + 16x + 61 ) + 249
2. Synthetic Division (cont..)
P(x) = D(x)Q(x) + R
dividend = divisor x quotient + remainder
3.4 Remainder Theorem and Factor
Theorem
1. Remainder Theorem
The remainder obtained in the synthetic division process has an
important interpretation, as described below
The Remainder Theorem tells you that synthetic division can be
used to evaluate a polynomial function.
That is, to evaluate a polynomial function f(x) when x = k,
divide f(x) by x – k. The remainder will be f(k)
Remainder Theorem
If a polynomial f(x) is divided by x – k , the remainder
is
r = f(k)
5 3611
3315
Example:
Use the Remainder Theorem to evaluate the following function at
Using synthetic division, you obtain the following
Because the remainder is r = 36, you can conclude that f(3) = 36.
This means that (3,36) is a point on the graph of f(x). You can check
this by substituting x=3 into original solution.
3 5 –4 3
Example:
f(x) = 5x2 – 4x + 3
f(3) = 5(3)2 – 4(3) + 3
f(3) = 5 ∙ 9 – 12 + 3
f(3) = 45 – 12 + 3
f(3) = 36
Check!
Notice that the value obtained when evaluating the
function at f(3) and the value of the remainder when
dividing the polynomial by x – 3 are the same.
2. Factor Theorem
• Another important theorem is the Factor Theorem. This theorem
states that you can test to see whether a polynomial has (x – k)
as a factor by evaluating the polynomial at x = k. If the result
is 0, (x – k) is a factor.
If the remainder is zero, that means f(k) = 0. This also means
that the divisor resulting in a remainder of zero is a factor of
the polynomial.
Factor Theorem
A polynomial f (x) has a factor (x – k)
if and only if f (k) = 0
Show that (x – 3) is factors of f(x) = x3 + 4x2 – 15x – 18. Then find
the remaining factors of f(x)
3 1 4 –15 –18
3 21 18
1 7 6 0
Using synthetic division, you obtain the following
Then function f(x) can be written as
f(x) = (x – 3)( x2 + 7x + 6 )
f(x) = (x – 3)(x + 6)(x + 1)
Therefore, (x – 3) is factors of f(x) = x3 + 4x2 – 15x – 18.
Remaining factors of f(x) are (x + 6)(x + 1)
Since the remainder is zero, f(3) =
0, show that (x – 3) is a factor of
x3 + 4x2 – 15x – 18.
Factoring x2 + 7x +6
x2 + 7x +6 = (x + 6)(x + 1)
Example:
The Factor Theorem
(x – 3)(x + 6)(x + 1).
Compare the factors
of the polynomials
to the zeros as seen
on the graph of
x3 + 4x2 – 15x – 18.
Example:
Given a polynomial and one of its factors, find the remaining
factors of the polynomial.
x3 – 11x2 + 14x + 80
x – 8
Ans:
(x – 8)(x – 5)(x + 2)
Factoring
Factoring is the breaking apart of a polynomial into
a product of other smaller polynomials. If you
choose, you could then multiply these factors
together, and you should get the original
polynomial.
For example, one set of factors for 24 is 6 and 4
because 6 times 4 = 24.
Root/Zero of a polynomial
A root or zero of a function is
a number that, when plugged in for the variable, makes
the function equal to zero. Thus, the roots of a
polynomial P(x) are values of x such that P(x) = 0.
3. The Rational Zero Test
The Rational Zero Theorem:
Suppose P(x) = anxn + an-1x
n-1 + an-2xn-2+ …+ a2x
2 + a1x + a0
is a polynomial with integer coefficients, and
Rational zero = p/q is a rational zero of P(x).
Then p is a factor of constant term, q is a factor of leading
coefficient.
Find the rational zeros of f(x) =1x3 - 3x2 - 2x + 6
The constant term, p = 6 and leading
coefficient, q = 1
p = {1, 2, 3, 6}
q = {1}
Possible rational-zeros:
p/q = {1, 2, 3, 6}
f(-1) = (-1)3 - 3(-1)2 -2(-1) + 6 = 4
f(1) = (1)3 - 3(1)2 -2(1) + 6 = 2
f(-2) = (-2)3 - 3(-2)2 -2(-2) + 6 = -10
f(2) = (2)3 - 3(2)2 -2(2) + 6 = -2
f(-3) = (-3)3 - 3(-3)2 -2(-3) + 6 = -42
f(3) = (3)3 - 3(3)2 -2(3) + 6 = 0
List all rational numbers whose numerators
are factors of the constant terms and whose
denominators are factors of the leading
coefficient.
List all the possible rational zeros
Use trial-and-error method to determine which ,
if any, are actual zeros of the polynomial.
(x - 3) is a factor, gives that x = 3 is a
rational zero
f(x) =x3 - 3x2 - 2x + 6 = (x - 3)(…………….…)
Cont. Find the rational zeros of f(x) =x3 - 3x2 - 2x + 6
Now find the remaining factor
Applying synthetic division
Rewrite the polynomial by including the factor
Factor of x2 – 2
produce x = - 2 and x = 2
f(x) = x3 - 3x2 - 2x + 6 = (x - 3)(…………….…)?
x3 - 3x2 - 2x + 6
1 -3 -2 6
3 0 -6
1 0 -2 0
x2 - 2
3
f(x) = x3 - 3x2 - 2x + 6 = (x - 3)(x2 - 2)
f(x) = (x - 3)( x - 2)(x + 2)
We conclude that the rational zeros of f are x = 3 , x = - 2 and x = 2
If a leading coefficient is not 1, the list of possible
rational zeros can increase dramatically.
E.g.
p = {1, 3}
q = {1, 2}
Possible rational zeros : p/q = {1, 3, 1
2,
3
2}
Rational Zero Test/Theorem can be used to find all
the rational zeros of a polynomial
Rational Zero Test often useful when we need to
factorize a polynomial with degree higher than 2!
1.5 Solving Quadratic Equations
In this section, we will learn 4 methods for solving
quadratic equations:
i) Factoring
ii) Extracting square roots
iii) Completing square
iv) The Quadratic formula.
49
(i) Factoring
Some quadratic equations can be solved by factoring and
using basic property of real numbers called Zero-Factor
Property.
Be sure you see that Zero-Factor Property works only for
equations written in general form (in which the right side of
the equation is zero). So, all terms must be collected on one
side before factoring.
To use this property, write the left hand side of the general
form of quadratic equation as the product of two linear
factors.
Then find the solutions of the quadratic equation by setting
each linear factor equal to zero.
50
(iii) Completing the Square
… is to write a quadratic function as a perfect square. Here are
some examples of perfect squares!
x2 + 6x + 9
x2 - 10x + 25
x2 + 12x + 36
Try to factor these (they’re easy).
Perfect Square Trinomials
x2 + 6x + 9
x2 - 10x + 25
x2 + 12x + 36
Can you see a numerical connection between …
6 and 9 using 3
-10 and 25 using -5
12 and 36 using 6
=(x+3)2
=(x-5)2
=(x+6)2
For a perfect square, the following relationships will always be
true …
x2 + 6x + 9
x2 - 10x + 25
The Perfect Square Connection
Half of these values squared … are these values
The Perfect Square Connection
In the following perfect square trinomial, the constant term is missing. Can you predict what it might be?
X2 + 14x + ____
Find the constant term by squaring half the coefficient of the linear term.
(14/2)2
X2 + 14x + 49
Perfect Square Trinomials
Create perfect square
trinomials.
x2 + 20x + ___
x2 - 4x + ___
x2 + 5x + ___
100
4
25/4
Solve the following equation by completing the square:
Step 1: Move the constant term (i.e. the number) to right side of the equation
2 8 20 0x x+ − =
2 8 20x x+ =
Solving Quadratic Equations by
Completing the Square
Step 2: Find the term that
completes the square on the left
side of the equation. Add that
term to both sides.
2 8 =20 + x x+ +
21( ) 4 then square it, 4 16
28• = =
2 8 2016 16x x+ + = +
Solving Quadratic Equations by
Completing the Square
Step 3: Factor the
perfect square
trinomial on the left
side of the
equation. Simplify
the right side of the
equation.
2 8 2016 16x x+ + = +
2
( 4)( 4) 36
( 4) 36
x x
x
− − =
− =
Solving Quadratic Equations by
Completing the Square
Solve this by
extracting the square root𝑥 + 4 = ± 36
𝑥 = −4 ± 36
Solving Quadratic Equations by
Completing the Square
Step 4: Set up
the two
possibilities
and solve
4 6
4 6 an
d 4 6
10 and 2 x=
x
x x
x
= −
=
= −
− − = − +
Completing the Square-Example #2
Solve the following equation by
completing the square:
Step 1: Move the constant to
the right side of the equation.
22 7 12 0x x− + =
22 7 12x x− = −
Solving Quadratic Equations by
Completing the Square
Step 2: Find the term that
completes the square on the left
side of the equation. Add that
term to both sides.
The quadratic coefficient must be
equal to 1 before you complete
the square, so you must divide
all terms by the quadratic
coefficient first.
2
2
2
2 7
2
2 2 2
7 12
7
2
=-12 +
6
x x
x x
xx
− +
− +
− +
= − +
= − +
21 7 7 49
( ) then square it, 2 62 4 4 1
7 • = =
2 49 49
16 1
76
2 6x x− + = − +
Solving Quadratic Equations by
Completing the Square
Step 3: Factor the
perfect square
trinomial on the left
side of the
equation. Simplify
the right side of the
equation.
2
2
2
76
2
7 96 49
4 16 16
7 47
4
49 49
16 1
16
6x x
x
x
− + = − +
− = − +
− = −
Use calculator to do this!Use calculator to do this!Use calculator to do this!Use calculator to do this!Use calculator to do this!
Linear Inequality74
An inequality in one variable is a linear inequality if
it can be written in one of the following forms.
The solution set can be written as set notation, (or
represented as interval notation or graph)
𝑎𝑥 + 𝑏 ≤ 0 𝑎𝑥 + 𝑏 < 0 𝑎𝑥 + 𝑏 ≥ 0 𝑎𝑥 + 𝑏 > 0
Solving Quadratic Inequality77
Example: Solve the inequality 𝑥2 − 5x < 0.
First, find the critical numbers of 𝑥2 − 5x < 0 by adding the solutions of the
equation 𝑥2 − 5x = 0𝑥2 − 5x = 0𝑥 𝑥 − 5 = 0x = 0, x = 5
Test intervals are (−∞, 0), (0,5) 𝑎𝑛𝑑 (5,+∞)
Because the inequality is
satisfied by the middle
test only, we can
conclude that the
solution set in the
interval (0,5)
Critical numbers
Absolute Value Equation79
Consider the absolute value equation
𝑥 = 3
The only solutions are x=-3 and x=3, because these
are the two real numbers whose distance from zero is
3.
Recap!
For |x|=a,
The solution is given by
x=-a or x=a