For use with linear factors

Extra SectionSynthetic Division

Warm-upDivide.

(3x3 +2x2 − x +3) ÷ (x −3)

Warm-upDivide.

(3x3 +2x2 − x +3) ÷ (x −3)

x −3 3x3 +2x2 − x +3

Warm-upDivide.

(3x3 +2x2 − x +3) ÷ (x −3)

x −3 3x3 +2x2 − x +3

3x2

Warm-upDivide.

(3x3 +2x2 − x +3) ÷ (x −3)

x −3 3x3 +2x2 − x +3

3x2

−(3x3 −9x2 )

Warm-upDivide.

(3x3 +2x2 − x +3) ÷ (x −3)

x −3 3x3 +2x2 − x +3

3x2

−(3x3 −9x2 )

11x2 − x

Warm-upDivide.

(3x3 +2x2 − x +3) ÷ (x −3)

x −3 3x3 +2x2 − x +3

3x2

−(3x3 −9x2 )

11x2 − x

+11x

Warm-upDivide.

(3x3 +2x2 − x +3) ÷ (x −3)

x −3 3x3 +2x2 − x +3

3x2

−(3x3 −9x2 )

11x2 − x

+11x

−(11x2 −33x)

Warm-upDivide.

(3x3 +2x2 − x +3) ÷ (x −3)

x −3 3x3 +2x2 − x +3

3x2

−(3x3 −9x2 )

11x2 − x

+11x

−(11x2 −33x)

32x +3

Warm-upDivide.

(3x3 +2x2 − x +3) ÷ (x −3)

x −3 3x3 +2x2 − x +3

3x2

−(3x3 −9x2 )

11x2 − x

+11x

−(11x2 −33x)

32x +3

+32

Warm-upDivide.

(3x3 +2x2 − x +3) ÷ (x −3)

x −3 3x3 +2x2 − x +3

3x2

−(3x3 −9x2 )

11x2 − x

+11x

−(11x2 −33x)

32x +3

+32

−(32x −96)

Warm-upDivide.

(3x3 +2x2 − x +3) ÷ (x −3)

x −3 3x3 +2x2 − x +3

3x2

−(3x3 −9x2 )

11x2 − x

+11x

−(11x2 −33x)

32x +3

+32

−(32x −96)

99

Warm-upDivide.

(3x3 +2x2 − x +3) ÷ (x −3)

x −3 3x3 +2x2 − x +3

3x2

−(3x3 −9x2 )

11x2 − x

+11x

−(11x2 −33x)

32x +3

+32

−(32x −96)

99

3x2 + 11x +32,R :99

Rational Roots Theorem

Rational Roots Theorem

Let p be all factors of the leading coefficient and q be all factors of the

constant in any polynomial. Then p/q gives all possible roots of the

polynomial.

Synthetic Division

Synthetic Division

Another way to divide polynomials, without the use of variables

Synthetic Division

Another way to divide polynomials, without the use of variables

Only works if you’re dividing by a linear factor

Synthetic Division

Another way to divide polynomials, without the use of variables

Only works if you’re dividing by a linear factor

Allows for us to test whether a possible root is an actual zero

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

1 4 0 −3 0 1 0 5

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

1 4 0 −3 0 1 0 5

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

1 4 0 −3 0 1 0 5

4

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

1 4 0 −3 0 1 0 5

4

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

1 4 0 −3 0 1 0 5

44

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

1 4 0 −3 0 1 0 5

444

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

1 4 0 −3 0 1 0 5

444

4

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

1 4 0 −3 0 1 0 5

444

41

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

1 4 0 −3 0 1 0 5

444

41

1

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

1 4 0 −3 0 1 0 5

444

41

11

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

1 4 0 −3 0 1 0 5

444

41

11

1

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

1 4 0 −3 0 1 0 5

444

41

11

12

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

1 4 0 −3 0 1 0 5

444

41

11

12

2

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

1 4 0 −3 0 1 0 5

444

41

11

12

22

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

1 4 0 −3 0 1 0 5

444

41

11

12

22

2

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

1 4 0 −3 0 1 0 5

444

41

11

12

22

27

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

1 4 0 −3 0 1 0 5

444

41

11

12

22

27

Example 1Determine whether 1 is a root of

4x6 −3x4 + x2 + 5

1 4 0 −3 0 1 0 5

444

41

11

12

22

27

4x5 + 4x4 + x3 + x2 +2x +2,R :7

Example 2Use synthetic division to find the quotient and

remainder.

(4x3 − 7x2 − 11x + 5) ÷ (4x − 5)

Example 2Use synthetic division to find the quotient and

remainder.

(4x3 − 7x2 − 11x + 5) ÷ (4x − 5)

4x − 5→ x − 5

4

Example 2Use synthetic division to find the quotient and

remainder.

(4x3 − 7x2 − 11x + 5) ÷ (4x − 5)

54 4 −7 −11 5

4x − 5→ x − 5

4

Example 2Use synthetic division to find the quotient and

remainder.

(4x3 − 7x2 − 11x + 5) ÷ (4x − 5)

54 4 −7 −11 5

4x − 5→ x − 5

4

4

Example 2Use synthetic division to find the quotient and

remainder.

(4x3 − 7x2 − 11x + 5) ÷ (4x − 5)

54 4 −7 −11 5

4x − 5→ x − 5

4

45

Example 2Use synthetic division to find the quotient and

remainder.

(4x3 − 7x2 − 11x + 5) ÷ (4x − 5)

54 4 −7 −11 5

4x − 5→ x − 5

4

45-2

Example 2Use synthetic division to find the quotient and

remainder.

(4x3 − 7x2 − 11x + 5) ÷ (4x − 5)

54 4 −7 −11 5

4x − 5→ x − 5

4

45-2

−52

Example 2Use synthetic division to find the quotient and

remainder.

(4x3 − 7x2 − 11x + 5) ÷ (4x − 5)

54 4 −7 −11 5

4x − 5→ x − 5

4

45-2

−52

−272

Example 2Use synthetic division to find the quotient and

remainder.

(4x3 − 7x2 − 11x + 5) ÷ (4x − 5)

54 4 −7 −11 5

4x − 5→ x − 5

4

45-2

−52

−272

−1358

Example 2Use synthetic division to find the quotient and

remainder.

(4x3 − 7x2 − 11x + 5) ÷ (4x − 5)

54 4 −7 −11 5

4x − 5→ x − 5

4

45-2

−52

−272

−1358

−958

Example 2Use synthetic division to find the quotient and

remainder.

(4x3 − 7x2 − 11x + 5) ÷ (4x − 5)

54 4 −7 −11 5

4x − 5→ x − 5

4

45-2

−52

−272

−1358

−958

4x2 −2x − 272

,R :− 958

Example 3Use synthetic division to find the quotient and

remainder.

(6x3 − 16x2 + 17x −6) ÷ (3x −2)

Example 3Use synthetic division to find the quotient and

remainder.

3x −2→ x − 2

3

(6x3 − 16x2 + 17x −6) ÷ (3x −2)

Example 3Use synthetic division to find the quotient and

remainder.

3x −2→ x − 2

3

(6x3 − 16x2 + 17x −6) ÷ (3x −2)

23 6 −16 17 −6

Example 3Use synthetic division to find the quotient and

remainder.

3x −2→ x − 2

3

(6x3 − 16x2 + 17x −6) ÷ (3x −2)

23 6 −16 17 −6

6

Example 3Use synthetic division to find the quotient and

remainder.

3x −2→ x − 2

3

(6x3 − 16x2 + 17x −6) ÷ (3x −2)

23 6 −16 17 −6

6

4

Example 3Use synthetic division to find the quotient and

remainder.

3x −2→ x − 2

3

(6x3 − 16x2 + 17x −6) ÷ (3x −2)

23 6 −16 17 −6

6

4

-12

Example 3Use synthetic division to find the quotient and

remainder.

3x −2→ x − 2

3

(6x3 − 16x2 + 17x −6) ÷ (3x −2)

23 6 −16 17 −6

6

4

-12

-8

Example 3Use synthetic division to find the quotient and

remainder.

3x −2→ x − 2

3

(6x3 − 16x2 + 17x −6) ÷ (3x −2)

23 6 −16 17 −6

6

4

-12

-8

9

Example 3Use synthetic division to find the quotient and

remainder.

3x −2→ x − 2

3

(6x3 − 16x2 + 17x −6) ÷ (3x −2)

23 6 −16 17 −6

6

4

-12

-8

9

6

Example 3Use synthetic division to find the quotient and

remainder.

3x −2→ x − 2

3

(6x3 − 16x2 + 17x −6) ÷ (3x −2)

23 6 −16 17 −6

6

4

-12

-8

9

6

0

Example 3Use synthetic division to find the quotient and

remainder.

3x −2→ x − 2

3

(6x3 − 16x2 + 17x −6) ÷ (3x −2)

23 6 −16 17 −6

6

4

-12

-8

9

6

0

6x2 − 12x +9,R :0

Factoring a Quadratic

Factoring a Quadratic

Multiply a and c

Factoring a Quadratic

Multiply a and c

Factor ac into two factors that add up to b

Factoring a Quadratic

Multiply a and c

Factor ac into two factors that add up to b

Replace b with these two values

Factoring a Quadratic

Multiply a and c

Factor ac into two factors that add up to b

Replace b with these two values

Group first 2 and last 2 terms

Factoring a Quadratic

Multiply a and c

Factor ac into two factors that add up to b

Replace b with these two values

Group first 2 and last 2 terms

Factor out the GCF of each

Factoring a Quadratic

Multiply a and c

Factor ac into two factors that add up to b

Replace b with these two values

Group first 2 and last 2 terms

Factor out the GCF of each

Factors: (Stuff inside)(Stuff outside)

Example 4Factor.

a. 2x2 + x −6 b. 4x2 − 19x + 12

Example 4Factor.

a. 2x2 + x −6 b. 4x2 − 19x + 12

2i−6

Example 4Factor.

a. 2x2 + x −6 b. 4x2 − 19x + 12

2i−6 = −12

Example 4Factor.

a. 2x2 + x −6 b. 4x2 − 19x + 12

2i−6 = −12 = 4(−3)

Example 4Factor.

a. 2x2 + x −6 b. 4x2 − 19x + 12

2i−6 = −12 = 4(−3)

2x2 + 4x −3x −6

Example 4Factor.

a. 2x2 + x −6 b. 4x2 − 19x + 12

2i−6 = −12 = 4(−3)

2x2 + 4x −3x −6

(2x2 + 4x)+ (−3x −6)

Example 4Factor.

a. 2x2 + x −6 b. 4x2 − 19x + 12

2i−6 = −12 = 4(−3)

2x2 + 4x −3x −6

(2x2 + 4x)+ (−3x −6)

2x(x +2)−3(x +2)

Example 4Factor.

a. 2x2 + x −6 b. 4x2 − 19x + 12

2i−6 = −12 = 4(−3)

2x2 + 4x −3x −6

(2x2 + 4x)+ (−3x −6)

2x(x +2)−3(x +2)

(x +2)(2x −3)

Example 4Factor.

a. 2x2 + x −6 b. 4x2 − 19x + 12

2i−6 = −12 = 4(−3)

2x2 + 4x −3x −6

(2x2 + 4x)+ (−3x −6)

2x(x +2)−3(x +2)

(x +2)(2x −3)

4i12 = 48

Example 4Factor.

a. 2x2 + x −6 b. 4x2 − 19x + 12

2i−6 = −12 = 4(−3)

2x2 + 4x −3x −6

(2x2 + 4x)+ (−3x −6)

2x(x +2)−3(x +2)

(x +2)(2x −3)

4i12 = 48 = (−16)(−3)

Example 4Factor.

a. 2x2 + x −6 b. 4x2 − 19x + 12

2i−6 = −12 = 4(−3)

2x2 + 4x −3x −6

(2x2 + 4x)+ (−3x −6)

2x(x +2)−3(x +2)

(x +2)(2x −3)

4i12 = 48 = (−16)(−3)

4x2 − 16x −3x + 12

Example 4Factor.

a. 2x2 + x −6 b. 4x2 − 19x + 12

2i−6 = −12 = 4(−3)

2x2 + 4x −3x −6

(2x2 + 4x)+ (−3x −6)

2x(x +2)−3(x +2)

(x +2)(2x −3)

4i12 = 48 = (−16)(−3)

4x2 − 16x −3x + 12

(4x2 − 16x)+ (−3x + 12)

Example 4Factor.

a. 2x2 + x −6 b. 4x2 − 19x + 12

2i−6 = −12 = 4(−3)

2x2 + 4x −3x −6

(2x2 + 4x)+ (−3x −6)

2x(x +2)−3(x +2)

(x +2)(2x −3)

4i12 = 48 = (−16)(−3)

4x2 − 16x −3x + 12

(4x2 − 16x)+ (−3x + 12)

4x(x − 4)−3(x − 4)

Example 4Factor.

a. 2x2 + x −6 b. 4x2 − 19x + 12

2i−6 = −12 = 4(−3)

2x2 + 4x −3x −6

(2x2 + 4x)+ (−3x −6)

2x(x +2)−3(x +2)

(x +2)(2x −3)

4i12 = 48 = (−16)(−3)

4x2 − 16x −3x + 12

(4x2 − 16x)+ (−3x + 12)

4x(x − 4)−3(x − 4)

(x − 4)(4x −3)

Homework

Homework

Worksheet!