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Synthetic Division
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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!