Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a...

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Section 4-3 The Remainder and Factor Theorems

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Transcript of Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a...

Page 1: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,

Section 4-3

The Remainder and Factor Theorems

Page 2: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,

Remainder Theorem

Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant, P(r), and where Q(x) is a polynomial with degree one less than the degree of P(x).

Page 3: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,

Example 1

Page 4: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,

Synthetic Divisionby x+3

62 xxRemainder 13

534 23 xxx

Page 5: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,
Page 6: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,

Divide using synthetic division.

yyyy 234 246 By y+2

Page 7: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,

Divide using synthetic division.

yyyy 234 246 By y+2

- 2 -1 0 4 0 3 2 0 2 -4 0 0 -6 8 -1 2 0 0 3 -4 8

432 45 yyy Remainder of 8

Page 8: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,

Divide using synthetic division. By (x+2)8102 xx

Page 9: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,

Divide using synthetic division. By (x-1)143 23 xxx

Page 10: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,

Divide using synthetic division. By (x+1)533 xx

Page 11: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,

Divide using synthetic division. By (x-4)4372 234 xxxx

Page 12: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,

Example 3

Page 13: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,

Example 4

Page 14: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,

Use the remainder and factor theorem to find the remainder for each division. State whether

the binomial is a factor of the polynomialby (x+1)

822 xx1624 23 xxx

Page 15: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,

Use the remainder and factor theorem to find the remainder for each division. State whether

the binomial is a factor of the polynomialby (x+4)

822 xx8102 xx

Page 16: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,

Use the remainder and factor theorem to find the remainder for each division. State whether

the binomial is a factor of the polynomialby (x-1)

822 xx123 x

Page 17: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,

Use the remainder and factor theorem to find the remainder for each division. State whether

the binomial is a factor of the polynomialby (x-4)

822 xx164 24 xx

Page 18: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,
Page 19: Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,

Example 5


The History of The Chinese Remainder Theorem · the Chinese Remainder Theorem, for the reason that it first appeared in a Chinese mathematical treatise. The Chinese Remainder Theorem

The History of The Chinese Remainder Theorem · the Chinese Remainder Theorem, for the reason that it first appeared in a Chinese mathematical treatise. The Chinese Remainder Theorem

Robustness in Chinese Remainder Theorem - arXiv · Robustness in Chinese Remainder Theorem Hanshen Xiao, Yufeng Huang, Yu Ye and Guoqiang Xiao Abstract—Chinese Remainder Theorem

Robustness in Chinese Remainder Theorem - arXiv · Robustness in Chinese Remainder Theorem Hanshen Xiao, Yufeng Huang, Yu Ye and Guoqiang Xiao Abstract—Chinese Remainder Theorem

Elementary Hydrostatics - The Maths Mann...Remainder theorem: if a polynomial f(.r) is divided by PX + q, the remainder is if divided by x — a, the remainder is f(a). Factor theorem:

Elementary Hydrostatics - The Maths Mann...Remainder theorem: if a polynomial f(.r) is divided by PX + q, the remainder is if divided by x — a, the remainder is f(a). Factor theorem:

Chinese Remainder Theorem

Chinese Remainder Theorem

Polynomials and Polynomial Functions - Andrews Universityrwright/algebra2/powerpoints/05...Remainder Theorem if polynomial f(x) is divided by the binomial (x –a), then the remainder

Polynomials and Polynomial Functions - Andrews Universityrwright/algebra2/powerpoints/05...Remainder Theorem if polynomial f(x) is divided by the binomial (x –a), then the remainder

Explain how the Remainder Theorem and the Factor Theorem … ·  · 2015-09-01Explain how the Remainder Theorem and the Factor ... examples: l. What is the remainder when f(x) by

Explain how the Remainder Theorem and the Factor Theorem … ·  · 2015-09-01Explain how the Remainder Theorem and the Factor ... examples: l. What is the remainder when f(x) by

33 taylor's remainder theorem i

33 taylor's remainder theorem i

REMAINDER AND FACTOR THEOREM… · Find the Remainder by using the Factor Theorem . 14x3 – 12x2 – 5x + 3 by x – 1 Find the Remainder by using the Factor Theorem . When 2x3 –

REMAINDER AND FACTOR THEOREM… · Find the Remainder by using the Factor Theorem . 14x3 – 12x2 – 5x + 3 by x – 1 Find the Remainder by using the Factor Theorem . When 2x3 –

The Remainder and Factor Theorems. The Remainder Theorem If a polynomial f(x) is divided by (x – a), the remainder is the constant f(a), and f(x) = q(x)

The Remainder and Factor Theorems. The Remainder Theorem If a polynomial f(x) is divided by (x – a), the remainder is the constant f(a), and f(x) = q(x)

1.2 – Factoring Polynomial Using the Remainder Theorem · 2019-11-05 · The Remainder Theorem: When Polynomial P(x) is divided by divisor (x – a), the remainder will be _____.

1.2 – Factoring Polynomial Using the Remainder Theorem · 2019-11-05 · The Remainder Theorem: When Polynomial P(x) is divided by divisor (x – a), the remainder will be _____.

Teacher Road Map for Lesson 21: The Remainder Theorem ·  · 2017-07-16Teacher Road Map for Lesson 21: The Remainder Theorem . ... Examples: 𝑔𝑔(𝑥𝑥 ... In order for x

Teacher Road Map for Lesson 21: The Remainder Theorem ·  · 2017-07-16Teacher Road Map for Lesson 21: The Remainder Theorem . ... Examples: 𝑔𝑔(𝑥𝑥 ... In order for x

Remainder and Factor Theorem Day 1 - Mrs. Allison's BLOG...Remainder Theorem: If a polynomial f (x) is divided by x — k , then the remainder, r , is r f (This connects synthetic

Remainder and Factor Theorem Day 1 - Mrs. Allison's BLOG...Remainder Theorem: If a polynomial f (x) is divided by x — k , then the remainder, r , is r f (This connects synthetic

The remainder theorem states that the remainder that you get when you divide a polynomial P(x) by (x – a) is equal to P(a). The factor theorem is.

The remainder theorem states that the remainder that you get when you divide a polynomial P(x) by (x – a) is equal to P(a). The factor theorem is.

Remainder and Factor Theorem F19 - Mrs. Allison's BLOG · Remainder Theorem: If a polynomial f (x) is divided by x — k , then the remainder, r , is r f (This connects synthetic

Remainder and Factor Theorem F19 - Mrs. Allison's BLOG · Remainder Theorem: If a polynomial f (x) is divided by x — k , then the remainder, r , is r f (This connects synthetic

The Chinese Remainder Theorem

The Chinese Remainder Theorem

Remainder theorem and factorization of polynomials

Remainder theorem and factorization of polynomials

X35 taylor's remainder theorem i

X35 taylor's remainder theorem i

Chinese remainder theorem and its applications

Chinese remainder theorem and its applications

Chinese Remainder Theorem. How many people What is x? Divided into 4s: remainder 3 x ≡ 3 (mod 4) Divided into 5s: remainder 4 x ≡ 4 (mod 5) Chinese Remainder.

Chinese Remainder Theorem. How many people What is x? Divided into 4s: remainder 3 x ≡ 3 (mod 4) Divided into 5s: remainder 4 x ≡ 4 (mod 5) Chinese Remainder.

Factor/Remainder theorem and quadratic theory - … ·  · 2015-02-02Factor/Remainder theorem and quadratic theory ... Examples are 3x3,3x3-1,3x2 + 2x, ... (x + 1) is a factor of

Factor/Remainder theorem and quadratic theory - … ·  · 2015-02-02Factor/Remainder theorem and quadratic theory ... Examples are 3x3,3x3-1,3x2 + 2x, ... (x + 1) is a factor of