Chapter 5: Polynomials and Factoring
Lecture notes
Math 1010
Section 5.1: Integers Exponents and Scientific Notation
Rules of Exponents
Let m and n be integers, and let a and b represent real numbers, variables, or algebraic expressions, a 6= 0,
b 6= 0.
(1) am · an = am+n
m
(2) aan = am−n
(3) (ab)m = am · bm
m n
mn
(4) (a
) = a
(5)
a
b
0
m
=
am
bm
(6) a = 1
(7) a−m = a1m
−m m
(8) ab
= ab
Ex.1
Simplify the following expressions:
(1) (x2 y 4 )(3x)
(2) (−2y 2 )3
5 3
b
(3) 14a
2 b2
7a
2 3
(4) x2y
(5)
(6)
xn y 3n
x2 y
(2a2 b3 )2
a3 b2
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Chapter 5: Polynomials and Factoring
Lecture notes
Ex.2
Evaluate each expression.
(1) 30
(2) 3−2
−1
(3) 34
Ex.3
Assume x 6= 0, y 6= 0. Rewrite each expression using only positive exponents.
3
(1) x−2
(2) (3x)1 −2
−2
(3) − 7x
y2
(4)
(5)
12x2 y −4
6x−1 y 2
3xy
x2 (5y)0
2
Math 1010
Chapter 5: Polynomials and Factoring
Lecture notes
Math 1010
Large and small numbers
Working with large and small numbers is much easier when we write them in a special format called
scientific notation.
Scientific notation
Scientific notation is a format in which a number is expressed as a number between 1 and 10 multiplied by a
power of 10.
Ex.4
one billion = 109 (ten to the ninth power)
6 billion
= 6 × 109
420
= 4.2 × 102
0.5
= 5 × 10−1
Ex.5
Write each number in scientific notation.
(1) 0.0000684
(2) 937, 200, 000
Ex.6
Write each number in decimal form.
(1) 6.45 × 10−2
(2) 3.42785 × 103
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Chapter 5: Polynomials and Factoring
Lecture notes
Math 1010
Ex.7
Rewrite the factors in scientific notation and then evaluate
(2, 400, 000, 000)(0.0000045)
(0.00003)(1500)
Section 5.2: Adding and Subtracting Polynomials
Definition of polynomial
Let an , an−1 , . . . , a0 be real numbers and let n be a non-negative integer. A polynomial in x is an expression
of the form
an xn + an−1 xn−1 + · · · + a1 x + a0
where an 6= 0. The polynomial is of degree n, and the number an is the leading coefficient. The number a0 is
the constant term. In the term ak xk , ak is the coefficient and k is the degree of the term.
A polynomial that is written in order of descending powers of the variable is said to be in standard form.
A polynomial with one term is called monomial. A polynomial with two unlike terms is called binomial and
a polynomial with three unlike terms is called trinomial.
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Chapter 5: Polynomials and Factoring
Lecture notes
Ex.1
Simplify the following polynomials
(1) (2x3 + x2 − 5) + (x2 + x + 6)
(2) (3x2 + 2x + 4) + (3x2 − 6x + 3) + (−x2 + 2x − 4)
(3) (5x3 + 2x2 − x + 7) + (3x2 − 4x + 7) + (−x3 + 4x2 − 8)
(4) (3x3 − 5x2 + 3) − (x3 + 2x2 − x − 4)
(5) (4x4 − 2x3 + 5x2 − x + 8) − (3x4 − 2x3 + 3x − 4)
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Math 1010
Chapter 5: Polynomials and Factoring
Lecture notes
Ex.2
Simplify the following polynomials
(1) (2x2 − 7x + 2) − (4x2 + 5x − 1) + (−x2 + 4x + 4)
(2) (−x2 + 4x − 3) − [(4x2 − 3x + 8) − (−x2 + x + 7)]
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Math 1010
Chapter 5: Polynomials and Factoring
Lecture notes
Math 1010
Position function
The position function is a polynomial of the form
h(t) = −16t2 + v0 t + s0
where the height h is measured in feet and the time t is measured in seconds. This position function gives
the height (above the ground) of a free-falling object. The coefficients of t, v0 , is called the initial velocity of
the object, and the constant term s0 is called the initial height of the object. If the initial velocity is positive,
the object was projected upward at t = 0 (initial time). If the initial velocity is negative, the object was
projected downward at t = 0. If the initial velocity is zero, the object was dropped at t = 0.
Ex.3
An object is thrown downward from the 86th floor observatory at the Empire State Building, which is
1050 feet high. The initial velocity is −15 feet per second. Use the position function
h(t) = −16t2 + v0 t + s0
to find the height of the object when t = 1, t = 4, and t = 7.
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Chapter 5: Polynomials and Factoring
Lecture notes
Section 5.3: Multiplying Polynomials
Ex.1
Multiply the following polynomials
(1) (2x − 7)(3x)
(2) (−x)(5x2 − x)
(3) (x − 3)(x + 3)
(4) (3x + 4)(2x + 1)
(5) (4x2 − 3x − 1)(2x − 5)
(6) (4x2 + x − 2)(5 + 3x − x2 )
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Math 1010
Chapter 5: Polynomials and Factoring
Lecture notes
Ex.2
Simplify the following expression (3x − 2)(3x + 2).
Ex.3
Simplify the following expression (2x − 7)2 .
Ex.4
Simplify the following expression (x + 4)3 .
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Math 1010
Chapter 5: Polynomials and Factoring
Lecture notes
Section 5.4: Factoring by Grouping and Special Forms
Ex.1
Find the greatest common factor of 6x5 , 30x4 , and 12x3 .
Ex.2
Factor out the greatest common monomial factor from 12x2 y − 28xy 2 .
Ex.3
Factor the polynomial −3x2 + 12x − 18 in two ways:
(1) Factor out a 3.
(2) Factor out a −3.
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Math 1010
Chapter 5: Polynomials and Factoring
Lecture notes
Ex.4
Factor the following expressions
(1) 5x2 (6x − 5) − 2(6x − 5)
(2) x3 − 5x2 + x − 5
(3) 4x3 + 3x − 8x2 − 6
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Math 1010
Chapter 5: Polynomials and Factoring
Lecture notes
Math 1010
Difference of two squares
Let u and v be real numbers, variables, or algebraic expressions. Then the expression u2 − v 2 can be factored
as follows
u2 − v 2 = (u + v)(u − v)
Ex.5
Factor the following polynomials
(1) x2 − 16
(2) 49y 2 − 25x2
Ex.6
Factor the expression (x + 2)2 − 9.
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Chapter 5: Polynomials and Factoring
Lecture notes
Math 1010
Sum or difference of two cubes
Let u and v be real numbers, variables, or algebraic expressions. Then the expressions u3 + v 3 and u3 − v 3
can be factored as follows
u3 + v 3 = (u + v)(u2 − uv + v 2 )
u3 − v 3 = (u − v)(u2 + uv + v 2 )
Ex.7
Factor the following polynomials
(1) x3 − 125
(2) 8y 3 − 1
(3) y 3 − 27x3
Ex.8
Factor each polynomial completely
(1) 125x2 − 80
(2) x4 − y 4
(3) 81m4 − 1
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Chapter 5: Polynomials and Factoring
Lecture notes
Section 5.5: Factoring Trinomials
Perfect square trinomials
Let u and v be real numbers, variables, or algebraic expressions.
u2 + 2uv + v 2 = (u + v)2
u2 − 2uv + v 2 = (u − v)2
Ex.1
Factor the following trinomials
(1) x2 − 4x + 4
(2) 9x2 − 30xy + 25y 2
(3) 32y 3 + 48y 2 + 18y
Perfect square trinomials
To factor a trinomial of the type x2 + bx + c consider
(x + m)(x + n) = x2 + (m + n)x + mn = x2 + bx + c
then b = m + n and c = mn.
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Math 1010
Chapter 5: Polynomials and Factoring
Lecture notes
Ex.2
Factor the following trinomials
(1) x2 + 3x − 4
(2) x2 − 2x − 8
(3) x2 − 5x + 6
Ex.3
Factor x2 − 17x − 18.
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Math 1010
Chapter 5: Polynomials and Factoring
Lecture notes
Ex.4
Factor 4x2 + 5x − 6.
Ex.5
Factor the following trinomials
(1) 2x2 − x − 21
(2) 6x2 + 19x + 10
(3) 8x2 y − 60xy + 28y
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Math 1010
Chapter 5: Polynomials and Factoring
Lecture notes
Ex.6
Factor −3x2 + 16x + 35.
Ex.7
Factor 3x2 + 5x − 2 by grouping.
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Math 1010
Chapter 5: Polynomials and Factoring
Lecture notes
Ex.8
Factor each polynomial completely.
(1) 3x2 − 108
(2) 4x3 − 32x2 + 64x
(3) x3 − 3x2 − 4x + 12
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Math 1010
Chapter 5: Polynomials and Factoring
Lecture notes
Math 1010
Section 5.6: Solving Polynomial Equations by Factoring
Zero-factor property
Let a and b be real numbers, variables, or algebraic expressions. If a and b are factors such that ab = 0, then
a = 0 or b = 0.
Definition of quadratic equation
A quadratic equation is an equation that can be written in the general form
ax2 + bx + c = 0
where a, b, and c are real numbers with a 6= 0.
Ex.1
Solve the following quadratic equations
(1) x2 − x − 6 = 0
(2) 2x2 + 5x = 12
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Chapter 5: Polynomials and Factoring
Lecture notes
Ex.2
Solve
(1) x2 − 2x + 16 = 6x
(2) (x + 3)(x + 6) = 4
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Math 1010