remainder theorem and factorization of polynomials

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Introducing a New Product

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 remainder of the division of P(x) by (x-2) is equal to P(2)=323-22+5=25

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

Exercise

What is the remainder of the division of P(x)=-4x3-4x2+9x by (x+2)?Check your result using the synthetic division

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

Lets consider the polynomialP(x)=x3-3x2-2x+6.

If we plug in x=3 into the polynomial, we get:P(3)=33-332-23+6 P(3)=0

Then, 3 is a root of the polynomial P(x)

Root of a polynomial. Example

Note that x-3 is a factor of the polynomialP(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 a root of P(x)=x3-3x2-2x+6?

Exercise

In that case, ...............is a factor of P(x).

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