example 3 Break-Even

Chapter 6.4

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

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

2009 PBLPathways

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

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

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

a. Graph the function using a window representing up to 50 thousand units and find one x-intercept of the graph.

x

P(x)

(20, 0)

3 20 0.1 11 80 2000x x x

2009 PBLPathways

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

a. Graph the function using a window representing up to 50 thousand units and find one x-intercept of the graph.

x

P(x)

(20, 0)

3 20 0.1 11 80 2000x x x

2009 PBLPathways

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

a. Graph the function using a window representing up to 50 thousand units and find one x-intercept of the graph.

x

P(x)

3 20 0.1 11 80 2000x x x

2009 PBLPathways

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

a. Graph the function using a window representing up to 50 thousand units and find one x-intercept of the graph.

x

P(x)

(20, 0)

3 20 0.1 11 80 2000x x x

2009 PBLPathways

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.

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

b. Use synthetic division to find a quadratic factor of P(x) .

20 0.1 11 80 2000

2 180 2000

0.1 9 100 0

2009 PBLPathways

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

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

b. Use synthetic division to find a quadratic factor of P(x) .

2009 PBLPathways

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

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

b. Use synthetic division to find a quadratic factor of P(x) .

2009 PBLPathways

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

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

b. Use synthetic division to find a quadratic factor of P(x) .

Multiply

2009 PBLPathways

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

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

b. Use synthetic division to find a quadratic factor of P(x) .

2009 PBLPathways

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

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

b. Use synthetic division to find a quadratic factor of P(x) .

2009 PBLPathways

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

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

b. Use synthetic division to find a quadratic factor of P(x) .

2009 PBLPathways

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

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

b. Use synthetic division to find a quadratic factor of P(x) .

2009 PBLPathways

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

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

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

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 0

RemainderCoefficients of quotient

2009 PBLPathways

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

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.

3 2 20.1 11 80 2000 20 0.1 9 100x x x x x x

Remainder

20 0.1 11 80 2000

2 180 2000

0.1 9 100 0

Coefficients of quotient

2009 PBLPathways

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

c. Find all of the zeros of P(x) .

3 2

2

2

0.1 11 80 2000 0

20 0.1 9 100 0

20 0.1 90 1000 0

0.1 20 100 10 0

x x x

x x x

x x x

x x x

20 0 or 100 0 or 10 0

20 100 10

x x x

x x x

2009 PBLPathways

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

c. Find all of the zeros of P(x) .

3 2

2

2

0.1 11 80 2000 0

20 0.1 9 100 0

20 0.1 90 1000 0

0.1 20 100 10 0

x x x

x x x

x x x

x x x

20 0 or 100 0 or 10 0

20 100 10

x x x

x x x

2009 PBLPathways

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

c. Find all of the zeros of P(x) .

3 2

2

2

0.1 11 80 2000 0

20 0.1 9 100 0

20 0.1 90 1000 0

0.1 20 100 10 0

x x x

x x x

x x x

x x x

20 0 or 100 0 or 10 0

20 100 10

x x x

x x x

2009 PBLPathways

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

c. Find all of the zeros of P(x) .

3 2

2

2

0.1 11 80 2000 0

20 0.1 9 100 0

20 0.1 90 1000 0

0.1 20 100 10 0

x x x

x x x

x x x

x x x

20 0 or 100 0 or 10 0

20 100 10

x x x

x x x

2009 PBLPathways

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

c. Find all of the zeros of P(x) .

3 2

2

2

0.1 11 80 2000 0

20 0.1 9 100 0

20 0.1 90 1000 0

0.1 20 100 10 0

x x x

x x x

x x x

x x x

20 0 or 100 0 or 10 0

20 100 10

x x x

x x x

2009 PBLPathways

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

c. Find all of the zeros of P(x) .

3 2

2

2

0.1 11 80 2000 0

20 0.1 9 100 0

20 0.1 90 1000 0

0.1 20 100 10 0

x x x

x x x

x x x

x x x

20 0 or 100 0 or 10 0

20 100 10

x x x

x x x

2009 PBLPathways

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

d. Determine the levels of production that give break-even.

x

P(x)

(-10,0) (20,0) (100,0)

2009 PBLPathways

The weekly profit for a product is thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even,

3 2( ) 0.1 11 80 2000P x x x x

d. Determine the levels of production that give break-even.

x

P(x)

(-10,0) (20,0) (100,0)

Break-even points