Remainder theorem and
factorization of polynomials

by Suso
Remainder theorem
The remainder of the division of a
polynomial P(x) by (x-a) is P(a).
For example, if P(x)=3x3-2x+5, the ...
Exercise
What is the remainder of the division of
P(x)=5x4-3x3+3x2+x-2 by (x-3)?
Check your result using the synthetic div...
Exercise
What is the remainder of the division of
P(x)=-4x3-4x2+9x by (x+2)?
Check your result using the synthetic divisio...
Exercise
Is the division (8x3-4x+12)(x+1) exact?
Exercise
Is the division (2x5+3x4+2x3-4x2+12)(x-2)
exact?
Exercise
What is the remainder of the division
(2x35+3x24+12)(x+1)?
Root of a polynomial
A number a is called a root (or a zero) of a
polynomial P(x) if P(a)=0.
Root of a polynomial. Example
Let´s consider the polynomial
P(x)=x3-3x2-2x+6.
If we plug in x=3 into the polynomial, we ge...
Root of a polynomial. Example
Note that x-3 is a factor of the polynomial
P(x)=x3-3x2-2x+6.

(the division is exact)
Roots and factors of a polynomial
a is a root of a polynomial P(x)
↕
P(a)=0
↕
(x-a) is a factor of P(x)
Exercise
Is− 2 a root of P(x)=x3-3x2-2x+6?
Exercise
In that case, ...............is a factor of P(x).
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Remainder theorem and factorization of polynomials

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A few notes about the Remainder Theorem, examples of use and the factorization of polynomials

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Remainder theorem and factorization of polynomials

  1. 1. Remainder theorem and factorization of polynomials by Suso
  2. 2. Remainder theorem The remainder of the division of a polynomial P(x) by (x-a) is P(a). For example, if P(x)=3x3-2x+5, the remainder of the division of P(x) by (x-2) is equal to P(2)=3·23-2·2+5=25
  3. 3. Exercise What is the remainder of the division of P(x)=5x4-3x3+3x2+x-2 by (x-3)? Check your result using the synthetic division
  4. 4. Exercise What is the remainder of the division of P(x)=-4x3-4x2+9x by (x+2)? Check your result using the synthetic division
  5. 5. Exercise Is the division (8x3-4x+12)(x+1) exact?
  6. 6. Exercise Is the division (2x5+3x4+2x3-4x2+12)(x-2) exact?
  7. 7. Exercise What is the remainder of the division (2x35+3x24+12)(x+1)?
  8. 8. Root of a polynomial A number a is called a root (or a zero) of a polynomial P(x) if P(a)=0.
  9. 9. Root of a polynomial. Example Let´s consider the polynomial P(x)=x3-3x2-2x+6. If we plug in x=3 into the polynomial, we get: P(3)=33-3·32-2·3+6 ⇒ P(3)=0 Then, 3 is a root of the polynomial P(x)
  10. 10. Root of a polynomial. Example Note that x-3 is a factor of the polynomial P(x)=x3-3x2-2x+6. (the division is exact)
  11. 11. Roots and factors of a polynomial a is a root of a polynomial P(x) ↕ P(a)=0 ↕ (x-a) is a factor of P(x)
  12. 12. Exercise Is− 2 a root of P(x)=x3-3x2-2x+6?
  13. 13. Exercise In that case, ...............is a factor of P(x).

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