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Polynomials
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Table of Contents
·
·
·
·
·
·
·
·
Factors and GCF
Factoring out GCF's
Factoring Polynomial w/ a=1
Factoring Using Special Patterns
Factoring Trinomial a#1
Factoring 4 Term Polynomials
Mixed Factoring
Solving Equations by Factoring
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Factors
and
Greatest Common Factors
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Table of
Contents
Factors of 10
Factors
Unique
to 10
Factors of 15
Factors
Unique
to 15
Factors 10 and 15
have in common
Number
Bank
1
11
2
12
3
13
4
5
6
7
8
9
10
14
15
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16
17
18
19
20
What is the greatest common factor (GCF) of 10 and 15?
Factors of 12
Factors
Unique
to 12
Factors of 18
Factors
Unique
to 18
Factors 12 and 18
have in common
Number
Bank
1
11
2
12
3
13
4
5
6
7
8
9
10
14
15
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16
17
18
19
20
What is the greatest common factor (GCF) of 12 and 18?
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1
What is the GCF of 12 and 15?
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2
What is the GCF of 24 and 48?
3
What is the GCF of 72 and 54?
4
What is the GCF of 70 and 99?
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5
What is the GCF of 28, 56 and 42?
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Variables also have a GCF.
The GCF of variables is the variable(s) that is in each term raised to
the lowest exponent given.
Example: Find the GCF
and
and
and
6
and
What is the GCF of
A
B
C
D
and
and
and
?
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Factoring
Using the
Distributive Property
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Table of
Contents
The first step in factoring is to determine its greatest monomial factor. If
there is a greatest monomial factor other than 1, use the distributive
property to rewrite the given polynomial as the product of this greatest
monomial factor and a polynomial.
Example 1
a)
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Factor each polynomial.
6x4 - 15x3 + 3x2
Find the GCF
GCF: 3x2
Reduce each term of
the polynomial dividing
by the GCF
3x2 (2x2 - 5x + 1)
The first step in factoring is to determine its greatest monomial factor. If
there is a greatest monomial factor other than 1, use the distributive
property to rewrite the given polynomial as the product of this greatest
monomial factor and a polynomial.
Example 1
Factor each polynomial.
b) 4m3n - 7m2n2
Find the GCF
GCF: m2n
Reduce each term of
the polynomial dividing
by the GCF
m2n(4n - 7n)
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Sometimes the distributive property can be used to factor a
polynomial that is not in simplest form but has a common binomial
factor.
Example 2
a)
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Factor each polynomial.
y(y - 3) + 7(y - 3)
Find the GCF
GCF: y - 3
Reduce each term of
the polynomial dividing
by the GCF
(y - 3)(y + 7)
Sometimes the distributive property can be used to factor a
polynomial that is not in simplest form but has a common binomial
factor.
Example 2
b)
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Factor each polynomial.
Find the GCF
GCF:
Reduce each term of
the polynomial dividing
by the GCF
In working with common binomial factors, look for factors that are
opposites of each other.
For example:
(x - y) = - (y - x) because
x - y = x + (-y) = -y + x = -(y - x)
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10 True or False: y - 7 = -1( 7 + y)
True
False
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11 True or False: 8 - d = -1( d + 8)
True
False
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12 True or False: 8c - h = -1( -8c + h)
True
False
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13 True or False: -a - b = -1( a + b)
True
False
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14 If possible, Factor
A
B
C
D Already Simplified
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15 If possible, Factor
A
B
C
D Already Simplified
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16 If possible, Factor
A
B
C
D Already Simplified
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17 If possible, Factor
A
B
C
D Already Simplified
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18 If possible, Factor
A
B
C
D Already Simplified
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19 If possible, Factor
A
B
C
D Already Simplified
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Factoring Trinomials:
x2 + bx + c
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Contents
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A polynomial that can be simplified to the form
ax + bx + c
, where a ≠ 0, is called a quadratic polynomial.
Qu
a
dr
Lin
at
Co
ns
t
er ant
te
m.
er
rm
m.
.
ea
ic
t
rt
A quadratic polynomial in which b ≠ 0 and c ≠ 0 is called a
quadratic trinomial. If only b=0 or c=0 it is called a quadratic
binomial. If both b=0 and c=0 it is a quadratic monomial.
Examples: Choose all of the description that apply.
Cubic
Quadratic
Linear
Constant
Trinomial
Binomial
Monomial
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20 Choose all of the descriptions that apply to:
A
Quadratic
B
Linear
C
Constant
D
Trinomial
E
Binomial
F
Monomial
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22 Choose all of the descriptions that apply to:
A
Quadratic
B
Linear
C
Constant
D
Trinomial
E
Binomial
F
Monomial
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23 Choose all of the descriptions that apply to:
A
Quadratic
B
Linear
C
Constant
D
Trinomial
E
Binomial
F
Monomial
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Simplify.
1)
(x + 2)(x + 3) = _________________________
3)
(x + 1)(x - 5) = ________________________
2)
4)
(x - 4)(x - 1) = _________________________
(x + 6)(x - 2) = ________________________
Answer
Bank
x2 - 5x + 4
x2 - 4x - 5
x2 + 5x + 6
x2 + 4x - 12
RECALL … What did we do?? Look for a pattern!!
Slide each polynomial
from the circle to the
correct expression.
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Examples:
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24 The factors of 12 will have what kind of signs given the
following equation?
A
Both positive
B
Both Negative
C
Bigger factor positive, the other negative
D
The bigger factor negative, the other positive
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25 The factors of 12 will have what kind of signs given the
following equation?
A
Both positive
B
Both negative
C
Bigger factor positive, the other negative
D
The bigger factor negative, the other positive
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26 Factor
A
(x + 12)(x + 1)
B
(x + 6)(x + 2)
C
(x + 4)(x + 3)
D
(x - 12)(x - 1)
E
(x - 6)(x - 1)
F
(x - 4)(x - 3)
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27 Factor
A
(x + 12)(x + 1)
B
(x + 6)(x + 2)
C
(x + 4)(x + 3)
D
(x - 12)(x - 1)
E
(x - 6)(x - 1)
F
(x - 4)(x - 3)
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28 Factor
A
(x + 12)(x + 1)
B
(x + 6)(x + 2)
C
(x + 4)(x + 3)
D
(x - 12)(x - 1)
E
(x - 6)(x - 1)
F
(x - 4)(x - 3)
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29 Factor
A
(x + 12)(x + 1)
B
(x + 6)(x + 2)
C
(x + 4)(x + 3)
D
(x - 12)(x - 1)
E
(x - 6)(x - 1)
F
(x - 4)(x - 3)
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Examples
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30 The factors of -12 will have what kind of signs given the
following equation?
A
Both positive
B
Both negative
C
Bigger factor positive, the other negative
D
The bigger factor negative, the other positive
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31 The factors of -12 will have what kind of signs given the
following equation?
A
Both positive
B
Both negative
C
Bigger factor positive, the other negative
D
The bigger factor negative, the other positive
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32 Factor
A
(x + 12)(x - 1)
B
(x + 6)(x - 2)
C
(x + 4)(x - 3)
D
(x - 12)(x + 1)
E
(x - 6)(x + 1)
F
(x - 4)(x + 3)
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33 Factor
A
(x + 12)(x - 1)
B
(x + 6)(x - 2)
C
(x + 4)(x - 3)
D
(x - 12)(x + 1)
E
(x - 6)(x + 1)
F
(x - 4)(x + 3)
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34 Factor
A
(x + 12)(x - 1)
B
(x + 6)(x - 2)
C
(x + 4)(x - 3)
D
(x - 12)(x + 1)
E
(x - 6)(x + 1)
F
(x - 4)(x + 3)
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Mixed Factoring
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36 Factor the following
A
(x - 2)(x - 4)
B
(x + 2)(x + 4)
C
(x - 2)(x +4)
D
(x + 2)(x - 4)
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37 Factor the following
A
(x - 3)(x - 5)
B
(x + 3)(x + 5)
C
(x - 3)(x +5)
D
(x + 3)(x - 5)
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38 Factor the following
A
(x - 3)(x - 4)
B
(x + 3)(x + 4)
C
(x +2)(x +6)
D
(x + 1)(x+12)
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39 Factor the following
A
(x - 2)(x - 5)
B
(x + 2)(x + 5)
C
(x - 2)(x +5)
D
(x + 2)(x - 5)
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Factor:
Factor out
STEP
1
STEP
2
STEP
3
STEP
4
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40 Factor completely:
A
B
C
D
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41 Factor completely:
A
B
C
D
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42 Factor completely:
A
B
C
D
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43 Factor completely:
A
B
C
D
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44 Factor completely:
A
B
C
D
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Factoring
Special Patterns
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Contents
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When we were multiplying polynomials we had special
patterns.
Square of Sums
Difference of Sums
Product of a Sum and a Difference
If we learn to recognize these squares and products we can use
them to help us factor.
Perfect Square Trinomials
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The Square of a Sum and the Square of a difference have products
that are called Perfect Square Trinomials.
How to Recognize a Perfect Square Trinomial:
Recall:
Observe the trinomial. The first term is a perfect square.
The second term is 2 times square root of
the first term times the square root of the
third. The sign is plus/minus.
The third term is a perfect square.
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Examples of Perfect Square Trinomials
Is the trinomial a perfect square?
Drag the Perfect Square
Trinomials into the Box.
Only Perfect Square Trinomials
will remain visible.
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45 Factor
A
B
C
D Not a perfect Square Trinomial
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46 Factor
A
B
C
D Not a perfect Square Trinomial
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47 Factor
A
B
C
D Not a perfect Square Trinomial
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Examples of Difference of Squares
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48 Factor
A
B
C
D
Not a Difference of Squares
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50 Factor
A
B
C
D
Not a Difference of Squares
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51 Factor
A
B
C
D
Not a Difference of Squares
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52 Factor using Difference of Squares:
A
B
C
D
Not a Difference of Squares
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Factoring Trinomials:
ax2 + bx + c
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How to factor a trinomial of the form ax² + bx + c.
Example:
Factor 2d² + 15d + 18
Find the product of a and c :
2 ∙ 18 = 36
Now find two integers whose product is 36 and whose sum is
equal to “b” or 15. Factors of 36
Sum = 15?
1 + 36 = 37
1, 36
2 + 18 = 20
2, 18
3 + 12 = 15
3, 12
Now substitute 12 + 3 into the equation for 15.
2d² + (12 + 3)d + 18
Distribute
2d² + 12d + 3d + 18
Group and factor GCF
2d(d + 6) + 3(d + 6)
Factor common binomial
(d + 6)(2d + 3)
Remember to check using FOIL!
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Factor.
15x² - 13x + 2
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a = 15 and c = 2, but b = -13
Since both a and c are positive, and b is negative we need
to find two negative factors of 30 that add up to -13
Factors of 30
-1, -30
-2, -15
-3, -10
-5, -6
Sum = -13?
-1 + -30 = -31
-2 + -15 = -17
-3 + -10 = -13
-5 + -6 = -11
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Factor
6y² - 13y - 5
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A polynomial that cannot be written as a product of two
polynomials is called a prime polynomial.
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53 Factor
A
B
C
D
Prime Polynomial
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54 Factor
A
B
C
D
Prime Polynomial
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55 Factor
A
B
C
D
Prime Polynomial
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Factoring 4 Term
Polynomials
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Polynomials with four terms like ab - 4b + 6a - 24, can
be factored by grouping terms of the polynomials.
Example 1:
ab - 4b + 6a - 24
(ab - 4b) + (6a - 24)
b(a - 4) + 6(a - 4)
(a - 4) (b + 6)
Group terms into binomials that can be
factored using the distributive property
Factor the GCF
Notice that a - 4 is a common binomial
factor and factor!
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Example 2:
6xy + 8x - 21y - 28
(6xy + 8x) + (-21y - 28)
2x(3y + 4) + (-7)(3y + 4)
(3y +4) (2x - 7)
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Group
Factor GCF
Factor common binomial
You must be able to recognize additive inverses!!!
(3 - a and a - 3 are additive inverses because their sum is equal to zero.)
Remember 3 - a = -1(a - 3).
Example 3:
15x - 3xy + 4y - 20
(15x - 3xy) + (4y - 20)
3x(5 - y) + 4(y - 5)
3x(-1)(-5 + y) + 4(y - 5)
-3x(y - 5) + 4(y - 5)
(y - 5) (-3x + 4)
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Group
Factor GCF
Notice additive inverses
Simplify
Factor common binomial
Remember to check each problem by using FOIL.
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56 Factor 15ab - 3a + 10b - 2
A
(5b - 1)(3a + 2)
B
(5b + 1)(3a + 2)
C
(5b - 1)(3a - 2)
D
(5b + 1)(3a - 1)
57 Factor 10m2n - 25mn + 6m - 15
A
(2m-5)(5mn-3)
B
(2m-5)(5mn+3)
C
(2m+5)(5mn-3)
D
(2m+5)(5mn+3)
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58 Factor 20ab - 35b - 63 +36a
A
(4a - 7)(5b - 9)
B
(4a - 7)(5b + 9)
C
(4a + 7)(5b - 9)
D
(4a + 7)(5b + 9)
59 Factor a2 - ab + 7b - 7a
A
(a - b)(a - 7)
B
(a - b)(a + 7)
C
(a + b)(a - 7)
D
(a + b)(a + 7)
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Mixed Factoring
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Contents
Summary of Factoring
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Factor the Polynomial
Factor out GCF
2 Terms
Difference
of Squares
4 Terms
3 Terms
Perfect Square
Trinomial
Factor the
Trinomial
a=1
Group and Factor
out GCF. Look for a
Common Binomial
a=1
Check each factor to see if it can be factored again.
If a polynomial cannot be factored, then it is called prime.
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60 Factor completely:
A
B
C
D
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61 Factor completely
A
B
C
D prime polynomial
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62 Factor
A
B
C
D
prime polynomial
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64 Factor
A
B
C
D
Prime Polynomial
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Solving Equations by
Factoring
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Table of
Contents
Given the following equation, what conclusion(s) can be
drawn?
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ab = 0
Since the product is 0, one of the factors, a or b, must be 0.
This is known as the Zero Product Property.
Given the following equation, what conclusion(s) can be drawn?
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(x - 4)(x + 3) = 0
Since the product is 0, one of the factors must be 0.
Therefore, either x - 4 = 0 or x + 3 = 0.
x - 4 = 0
+4 +4
x = 4
or
or
x + 3 = 0
-3 -3
x = -3
Therefore, our solution set is {-3, 4}. To verify the results, substitute each
solution back into the original equation.
To check x = 4: (x - 4)(x + 3) = 0
To check x = -3: (x - 4)(x + 3) = 0
(-3 - 4)(-3 + 3) = 0
(-7)(0) = 0
0=0
(4 - 4)(4 + 3) = 0
(0)(7) = 0
0=0
What if you were given the following equation?
How would you solve it?
We can use the Zero Product Property to solve it.
How can we turn this polynomial into a multiplication problem? Factor it!
Factoring yields:
(x - 6)(x + 4) = 0
By the Zero Product Property:
x-6=0
or
x+4=0
After solving each equation, we arrive at our solution:
{-4, 6}
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65 Choose all of the solutions to:
A
B
C
D
E
F
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66 Choose all of the solutions to:
A
B
C
D
E
F
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67 Choose all of the solutions to:
A
B
C
D
E
F
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68 A ball is thrown with its height at any time given by
When does the ball hit the ground?
-1 seconds
A
B
0 seconds
C
9 seconds
D
10 seconds