example 3
Break-Even
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
a. Graph the function using a window representing up to 50 thousand units and find
one x-intercept of the graph.
b. Use synthetic division to find a quadratic factor of P(x) .
c. Find all of the zeros of P(x) .
d. Determine the levels of production that give break-even.
Chapter 6.4
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
a. Graph the function using a window representing up to 50 thousand units and find
one x-intercept of the graph.
b. Use synthetic division to find a quadratic factor of P(x) .
c. Find all of the zeros of P(x) .
d. Determine the levels of production that give break-even.
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
a. Graph the function using a window representing up to 50 thousand units and find
one x-intercept of the graph.
0  0.1x3  11x 2  80 x  2000
P(x)
(20, 0)
x
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
a. Graph the function using a window representing up to 50 thousand units and find
one x-intercept of the graph.
0  0.1x3  11x 2  80 x  2000
P(x)
(20, 0)
x
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
a. Graph the function using a window representing up to 50 thousand units and find
one x-intercept of the graph.
0  0.1x3  11x 2  80 x  2000
P(x)
x
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
a. Graph the function using a window representing up to 50 thousand units and find
one x-intercept of the graph.
0  0.1x3  11x 2  80 x  2000
P(x)
(20, 0)
x
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
b. Use synthetic division to find a quadratic factor of P(x) .
1. Arrange the coefficients in descending powers of x, with a 0 for any
missing power. Place a from x - a to the left of the coefficients.
20 0.1  11  80  2000
 2  180  2000
 0.1  9  100 
0
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
b. Use synthetic division to find a quadratic factor of P(x) .
1. Arrange the coefficients in descending powers of x, with a 0 for any
missing power. Place a from x - a to the left of the coefficients.
20 0.1  11  80  2000
 2  180  2000
 0.1  9  100 
0
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
b. Use synthetic division to find a quadratic factor of P(x) .
2. Bring down the first coefficient to the third line. Multiply the last number
in the third line by a and write the product in the second line under the
next term.
20 0.1  11  80  2000
 2  180  2000
 0.1  9  100 
0
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
b. Use synthetic division to find a quadratic factor of P(x) .
2. Bring down the first coefficient to the third line. Multiply the last number
in the third line by a and write the product in the second line under the
next term.
20 0.1  11  80  2000
 2  180  2000
Multiply
 0.1  9  100 
0
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
b. Use synthetic division to find a quadratic factor of P(x) .
3. Add the last number in the second line to the number above it in the first
line. Continue this process until all numbers in the first line are used.
20 0.1  11  80  2000
 2  180  2000
 0.1  9  100 
0
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
b. Use synthetic division to find a quadratic factor of P(x) .
3. Add the last number in the second line to the number above it in the first
line. Continue this process until all numbers in the first line are used.
20 0.1  11  80  2000
 2  180  2000
 0.1  9  100 
0
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
b. Use synthetic division to find a quadratic factor of P(x) .
3. Add the last number in the second line to the number above it in the first
line. Continue this process until all numbers in the first line are used.
20 0.1  11  80  2000
 2  180  2000
 0.1  9  100 
0
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
b. Use synthetic division to find a quadratic factor of P(x) .
3. Add the last number in the second line to the number above it in the first
line. Continue this process until all numbers in the first line are used.
20 0.1  11  80  2000
 2  180  2000
 0.1  9  100 
0
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
b. Use synthetic division to find a quadratic factor of P(x) .
3. Add the last number in the second line to the number above it in the first
line. Continue this process until all numbers in the first line are used.
20 0.1  11  80  2000
 2  180  2000
 0.1  9  100 
0
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
b. Use synthetic division to find a quadratic factor of P(x) .
4. The third line represents the coefficients of the quotient, with the last
number the remainder. The quotient is a polynomial of degree one less
than the dividend.
20 0.1  11  80  2000
 2  180  2000
 0.1  9  100 
Coefficients of quotient
0
Remainder
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
b. Use synthetic division to find a quadratic factor of P(x) .
4. If the remainder is 0, x – a is a factor of the polynomial, and the
polynomial can be written as the product of the divisor x - a and the
quotient.
0.1x3  11x 2  80 x  2000   x  20   0.1x 2  9 x  100 
20 0.1  11  80  2000
 2  180  2000
 0.1  9  100 
Coefficients of quotient
0
Remainder
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
c. Find all of the zeros of P(x) .
0.1x 3  11x 2  80 x  2000  0
 x  20   0.1x 2  9 x  100   0
 x  20   0.1 x 2  90 x  1000   0
0.1 x  20  x  100  x  10   0
x  20  0 or
x  20
x  100  0
or
x  100
x  10  0
x  10
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
c. Find all of the zeros of P(x) .
0.1x 3  11x 2  80 x  2000  0
 x  20   0.1x 2  9 x  100   0
 x  20   0.1 x 2  90 x  1000   0
0.1 x  20  x  100  x  10   0
x  20  0 or
x  20
x  100  0
or
x  100
x  10  0
x  10
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
c. Find all of the zeros of P(x) .
0.1x 3  11x 2  80 x  2000  0
 x  20   0.1x 2  9 x  100   0
 x  20   0.1 x 2  90 x  1000   0
0.1 x  20  x  100  x  10   0
x  20  0 or
x  20
x  100  0
or
x  100
x  10  0
x  10
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
c. Find all of the zeros of P(x) .
0.1x 3  11x 2  80 x  2000  0
 x  20   0.1x 2  9 x  100   0
 x  20   0.1 x 2  90 x  1000   0
0.1 x  20  x  100  x  10   0
x  20  0 or
x  20
x  100  0
or
x  100
x  10  0
x  10
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
c. Find all of the zeros of P(x) .
0.1x 3  11x 2  80 x  2000  0
 x  20   0.1x 2  9 x  100   0
 x  20   0.1 x 2  90 x  1000   0
0.1 x  20  x  100  x  10   0
x  20  0 or
x  20
x  100  0
or
x  100
x  10  0
x  10
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
c. Find all of the zeros of P(x) .
0.1x 3  11x 2  80 x  2000  0
 x  20   0.1x 2  9 x  100   0
 x  20   0.1 x 2  90 x  1000   0
0.1 x  20  x  100  x  10   0
x  20  0 or
x  20
x  100  0
or
x  100
x  10  0
x  10
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
d. Determine the levels of production that give break-even.
P(x)
(-10,0)
(20,0)
(100,0)
x
2009 PBLPathways
3
2
The weekly profit for a product is P( x)  0.1x  11x  80 x  2000 thousand dollars,
where x is the number of thousands of units produced and sold. To find the number of
units that gives break-even,
d. Determine the levels of production that give break-even.
P(x)
(-10,0)
(20,0)
(100,0)
x
Break-even points
2009 PBLPathways