REATK4-002 Workbook

Quotient & Remainder for Algebraic Fractions

27 4 for example, then you get 6 as a quotient and 3 as the remainder . You 27 = 4 × 6 + 3 to show the connection between these values.

If you work out

can write this down as

The same thing applies in algebra when one polynomial is divided by another.

2 + 7 − 5 is divided by + for the quotient and as the remainder gives: 2 + 7 − 5 = ( − 4)( + ) + = 2 + ( − 4 ) + − 4

− 4 .

Example Find the quotient and remainder when

Using

2 gives:

Equating coefficients of = 1 Equating coefficients of (and using

= 1 from above) gives:

− 4 = 7 − 4 = −5

− 4 = 7 − 44 = −5

= 11 = 39

= 11 from above) gives:

Equating the constant terms (and using

Substituting for a, b and c gives: 2 + 7 − 5 = ( − 4)( + 11) + 39 In this example, the quotient and remainder of ( 2 + 7 − 5 = ( − 4)( + ) + and equating coefficients. Another way to use this identity is to substitute particular values of . It will be seen that when = 4 , the RHS simplifies to and so the remainder is easily found by substituting = 4 on the LHS. This gives = 4 2 + 7 × 4 − 5 as before. In its generalised form this result is known as the remainder theorem which states: When a polynomial ( ) is divided by ( − ) the remainder is ( ) 2 + 7 − 5) ÷ ( − 4) were found by using

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