example 2 Combining Graphical and Algebraic Methods

Chapter 6.4

Solve the equation .3 29 610 600 0x x x

2009 PBLPathways

Solve the equation .3 29 610 600 0x x x

2009 PBLPathways

Solve the equation .3 29 610 600 0x x x

3 2( ) 9 610 600P x x x x

x

P(x)

2009 PBLPathways

Solve the equation .3 29 610 600 0x x x

(1,0)

3 2( ) 9 610 600P x x x x

x

P(x)

2009 PBLPathways

Solve the equation .3 29 610 600 0x x x

(1,0)

3 2(1) 1 9 1 610 1 600 0P ?

3 2( ) 9 610 600P x x x x

x

P(x)

2009 PBLPathways

Solve the equation .3 29 610 600 0x x x

(1,0)

3 2(1) 1 9 1 610 1 600 0P

3 2( ) 9 610 600P x x x x

x

P(x)

2009 PBLPathways

Solve the equation .3 29 610 600 0x x x

(1,0)

3 2(1) 1 9 1 610 1 600 0P

3 2( ) 9 610 600P x x x x

x

P(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.

Solve the equation .3 29 610 600 0x x x

1 1 9 610 600

1 10 600

1 10 600 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.

Solve the equation .3 29 610 600 0x x x

1 1 9 610 600

1 10 600

1 10 600 0

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.

Solve the equation .3 29 610 600 0x x x

1 1 9 610 600

1 10 600

1 10 600 0

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.

Solve the equation .3 29 610 600 0x x x

1 1 9 610 600

1 10 600

1 10 600 0

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.

Solve the equation .3 29 610 600 0x x x

1 1 9 610 600

1 10 600

1 10 600 0

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.

Solve the equation .3 29 610 600 0x x x

1 1 9 610 600

1 10 600

1 10 600 0

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.

Solve the equation .3 29 610 600 0x x x

1 1 9 610 600

1 10 600

1 10 600 0

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.

Solve the equation .3 29 610 600 0x x x

1 1 9 610 600

1 10 600

1 10 600 0

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.

Solve the equation .3 29 610 600 0x x x

1 1 9 610 600

1 10 600

1 10 600 0

2009 PBLPathways

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.

Solve the equation .3 29 610 600 0x x x

1 1 9 610 600

1 10 600

1 10 600 0

Remainder

2009 PBLPathways

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.

Solve the equation .3 29 610 600 0x x x

1 1 9 610 600

1 10 600

1 10 600 0

3 2 29 610 600 1 10 600x x x x x x

Remainder

2009 PBLPathways

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.

Solve the equation .3 29 610 600 0x x x

1 1 9 610 600

1 10 600

1 10 600 0

3 2 29 610 600 1 10 600x x x x x x

Remainder

Divisor Quotient

2009 PBLPathways

Solve the equation .3 29 610 600 0x x x

3 2

2

9 610 600 0

1 10 600 0

x x x

x x x

21 0 or 10 600 01 30 20 0

x x xx x x

30 0 or 20 0

30 20

x x

x x

2009 PBLPathways

Solve the equation .3 29 610 600 0x x x

3 2

2

9 610 600 0

1 10 600 0

x x x

x x x

21 0 or 10 600 01 30 20 0

x x xx x x

30 0 or 20 0

30 20

x x

x x

2009 PBLPathways

Solve the equation .3 29 610 600 0x x x

3 2

2

9 610 600 0

1 10 600 0

x x x

x x x

21 0 or 10 600 01 30 20 0

x x xx x x

30 0 or 20 0

30 20

x x

x x

2009 PBLPathways

Solve the equation .3 29 610 600 0x x x

3 2

2

9 610 600 0

1 10 600 0

x x x

x x x

21 0 or 10 600 01 30 20 0

x x xx x x

30 0 or 20 0

30 20

x x

x x

2009 PBLPathways

Solve the equation .3 29 610 600 0x x x

3 2

2

9 610 600 0

1 10 600 0

x x x

x x x

21 0 or 10 600 01 30 20 0

x x xx x x

30 0 or 20 0

30 20

x x

x x

2009 PBLPathways

Solve the equation .3 29 610 600 0x x x

3 2

2

9 610 600 0

1 10 600 0

x x x

x x x

21 0 or 10 600 01 30 20 0

x x xx x x

30 0 or 20 0

30 20

x x

x x

2009 PBLPathways

Solve the equation .3 29 610 600 0x x x

3 2

2

9 610 600 0

1 10 600 0

x x x

x x x

21 0 or 10 600 01 30 20 0

x x xx x x

30 0 or 20 0

30 20

x x

x x

2009 PBLPathways

Solve the equation .3 29 610 600 0x x x

3 2( ) 9 610 600P x x x x

x

P(x)

2009 PBLPathways

Solve the equation .3 29 610 600 0x x x

(-30,0) (1, 0) (20, 0)

3 2( ) 9 610 600P x x x x

x

P(x)