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Warm Up
Warm Up
Without a calculator divide the following problems using long division.
a 218 ÷ 7
b 5361 ÷ 9
c 2712 ÷ 26
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 1 / 20
Polynomial and Synthetic Division
Pre-Calculus
Mr. Niedert
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 2 / 20
Polynomial and Synthetic Division
1 Long Division of Polynomials
2 Synthetic Division
3 The Remainder Theorem
4 The Factor Theorem
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 3 / 20
Polynomial and Synthetic Division
1 Long Division of Polynomials
2 Synthetic Division
3 The Remainder Theorem
4 The Factor Theorem
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 3 / 20
Polynomial and Synthetic Division
1 Long Division of Polynomials
2 Synthetic Division
3 The Remainder Theorem
4 The Factor Theorem
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 3 / 20
Polynomial and Synthetic Division
1 Long Division of Polynomials
2 Synthetic Division
3 The Remainder Theorem
4 The Factor Theorem
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 3 / 20
Purpose of Polynomial Long Division
Suppose that you are given the graph off (x) = 6x3 − 19x2 + 16x − 4, as seen below.
We can see that f has a zero at x = 2, but it is unclear where theother two zeros exist.
If we are able to remove (x − 2) as a factor then we are able to workwith a quadratic function, as opposed to a cubic, and we know howto find the zeros of any quadratic function.
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 4 / 20
Purpose of Polynomial Long Division
Suppose that you are given the graph off (x) = 6x3 − 19x2 + 16x − 4, as seen below.
We can see that f has a zero at x = 2, but it is unclear where theother two zeros exist.
If we are able to remove (x − 2) as a factor then we are able to workwith a quadratic function, as opposed to a cubic, and we know howto find the zeros of any quadratic function.
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 4 / 20
Purpose of Polynomial Long Division
Suppose that you are given the graph off (x) = 6x3 − 19x2 + 16x − 4, as seen below.
We can see that f has a zero at x = 2, but it is unclear where theother two zeros exist.
If we are able to remove (x − 2) as a factor then we are able to workwith a quadratic function, as opposed to a cubic, and we know howto find the zeros of any quadratic function.
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 4 / 20
Long Division of Polynomials
Example
Divide(6x3 − 19x2 + 16x − 4
)by (x − 2), and use the result to factor the
polynomial completely.
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 5 / 20
Long Division of Polynomials
Practice
Divide the following polynomials using polynomial long division. Use theresult to factor the polynomial completely.
a(5x2 − 17x − 12
)÷ (x − 4)
b(x4 − 1
)÷ (x + 1)
c(6x4 − x3 − x2 + 9x − 3
)÷(x2 + x − 1
)
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 6 / 20
Polynomial and Synthetic Division (Part 1 of 3)Assignment
pg. 159-160 Exercises #5-15 odd
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 7 / 20
Purpose of Synthetic Division
When dividing by divisors of the form x − k , we can use syntheticdivision as a shortcut for polynomial long division.
Keep in mind though that synthetic division works only for divisor ofthe form x − k . Specifically, you cannot use synthetic division todivide a polynomial by a quadratic polynomial (such as x2 + 2x + 1)or any other higher degree polynomial.
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 8 / 20
Purpose of Synthetic Division
When dividing by divisors of the form x − k , we can use syntheticdivision as a shortcut for polynomial long division.
Keep in mind though that synthetic division works only for divisor ofthe form x − k . Specifically, you cannot use synthetic division todivide a polynomial by a quadratic polynomial (such as x2 + 2x + 1)or any other higher degree polynomial.
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 8 / 20
Using Synthetic Division
Example
Use synthetic division to divide x4 − 10x2 − 2x + 4 by x + 3.
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 9 / 20
Using Synthetic Division
Practice
Use synthetic division to divide 5x3 + 8x2 − x + 6 by x + 2.
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 10 / 20
Two Common Errors with Synthetic Long Division toAvoid
1 Missing Powers of x
I If there are missing powers of x (as in the first example today), theremust be a placeholder in the synthetic division for each missing term.
I For example, x5 − 32 would be represented as 1 0 0 0 0 − 32.
2 Using AdditionI When doing long division we subtract the columns, but when using
synthetic division we add the columns.
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 11 / 20
Two Common Errors with Synthetic Long Division toAvoid
1 Missing Powers of xI If there are missing powers of x (as in the first example today), there
must be a placeholder in the synthetic division for each missing term.
I For example, x5 − 32 would be represented as 1 0 0 0 0 − 32.
2 Using AdditionI When doing long division we subtract the columns, but when using
synthetic division we add the columns.
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 11 / 20
Two Common Errors with Synthetic Long Division toAvoid
1 Missing Powers of xI If there are missing powers of x (as in the first example today), there
must be a placeholder in the synthetic division for each missing term.I For example, x5 − 32 would be represented as 1 0 0 0 0 − 32.
2 Using AdditionI When doing long division we subtract the columns, but when using
synthetic division we add the columns.
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 11 / 20
Two Common Errors with Synthetic Long Division toAvoid
1 Missing Powers of xI If there are missing powers of x (as in the first example today), there
must be a placeholder in the synthetic division for each missing term.I For example, x5 − 32 would be represented as 1 0 0 0 0 − 32.
2 Using Addition
I When doing long division we subtract the columns, but when usingsynthetic division we add the columns.
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 11 / 20
Two Common Errors with Synthetic Long Division toAvoid
1 Missing Powers of xI If there are missing powers of x (as in the first example today), there
must be a placeholder in the synthetic division for each missing term.I For example, x5 − 32 would be represented as 1 0 0 0 0 − 32.
2 Using AdditionI When doing long division we subtract the columns, but when using
synthetic division we add the columns.
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 11 / 20
Polynomial and Synthetic Division (Part 2 of 3)Assignment
pg. 159-160 Exercises #5-15 odd, 19-35 EOO
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 12 / 20
The Remainder Theorem
Theorem
If a polynomial f (x) is divided by x − k , the remainder is r = f (k).
The Remainder Theorem tells us that the synthetic division can beused to evaluate a polynomial function.
That is, to evaluate a polynomial function f (x) when x = k , dividef (x) by x − k .
The remainder will be f (k).
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 13 / 20
The Remainder Theorem
Theorem
If a polynomial f (x) is divided by x − k , the remainder is r = f (k).
The Remainder Theorem tells us that the synthetic division can beused to evaluate a polynomial function.
That is, to evaluate a polynomial function f (x) when x = k , dividef (x) by x − k .
The remainder will be f (k).
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 13 / 20
The Remainder Theorem
Theorem
If a polynomial f (x) is divided by x − k , the remainder is r = f (k).
The Remainder Theorem tells us that the synthetic division can beused to evaluate a polynomial function.
That is, to evaluate a polynomial function f (x) when x = k , dividef (x) by x − k .
The remainder will be f (k).
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 13 / 20
Using the Remainder Theorem
Example
Use the Remainder Theorem to evaluate the functionf (x) = 3x3 + 8x2 + 5x − 7 at x = −2.
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 14 / 20
Using the Remainder Theorem
Practice
Use the Remainder Theorem to evaluate the functionf (x) = 4x3 + 10x2 − 3x − 8 at the value f (−1).
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 15 / 20
The Factor Theorem
Theorem
A polynomial f (x) has a factor (x − k) if and only if f (k) = 0.
This theorem states that you can test to see whether a polynomialhas (x − k) as a factor by evaluating the polynomial at x = k.
If the result is 0, (x − k) is a factor.
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 16 / 20
The Factor Theorem
Theorem
A polynomial f (x) has a factor (x − k) if and only if f (k) = 0.
This theorem states that you can test to see whether a polynomialhas (x − k) as a factor by evaluating the polynomial at x = k.
If the result is 0, (x − k) is a factor.
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 16 / 20
Factoring a Polynomial: Repeated Division
Example
Show that (x − 2) and (x + 3) of f (x) = 2x4 + 7x3 − 4x2 − 27x − 18.Then find the remaining factors of f (x).
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 17 / 20
Factoring a Polynomial: Repeated Division
Practice
Show that (x + 3) is a factor of x3 − 19x − 30 = 0. Then find theremaining factors of f (x).
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 18 / 20
Uses of the Remainder of Synthetic Division
Uses of the Remainder of Synthetic Division
The remainder r , obtained in the synthetic division of f (x) by x − k ,provided the following information.
1 The remainder r gives the value of f at x = k . That is, r = f (k).
2 If r = 0, (x − k) is a factor of f (x).
3 If r = 0, (k, 0) is an x-intercept of the graph of f .
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 19 / 20
Uses of the Remainder of Synthetic Division
Uses of the Remainder of Synthetic Division
The remainder r , obtained in the synthetic division of f (x) by x − k ,provided the following information.
1 The remainder r gives the value of f at x = k . That is, r = f (k).
2 If r = 0, (x − k) is a factor of f (x).
3 If r = 0, (k, 0) is an x-intercept of the graph of f .
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 19 / 20
Uses of the Remainder of Synthetic Division
Uses of the Remainder of Synthetic Division
The remainder r , obtained in the synthetic division of f (x) by x − k ,provided the following information.
1 The remainder r gives the value of f at x = k . That is, r = f (k).
2 If r = 0, (x − k) is a factor of f (x).
3 If r = 0, (k, 0) is an x-intercept of the graph of f .
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 19 / 20
Uses of the Remainder of Synthetic Division
Uses of the Remainder of Synthetic Division
The remainder r , obtained in the synthetic division of f (x) by x − k ,provided the following information.
1 The remainder r gives the value of f at x = k . That is, r = f (k).
2 If r = 0, (x − k) is a factor of f (x).
3 If r = 0, (k, 0) is an x-intercept of the graph of f .
Pre-Calculus Polynomial and Synthetic Division Mr. Niedert 19 / 20
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