Factor & Remainder Theorem (DP IB Analysis & Approaches (AA)): Revision Note

Factor theorem

What is the factor theorem?

• The factor theorem states that for any polynomial function the following is true

  • If is a factor of then

  • If then is a factor of

• For example:

  • 

    • so is a factor of

  • 

    • so

• The factor theorem connects the roots of a polynomial equation and the factors of the polynomial function

  • Rearrange the root to make it equal to zero to find the factor

    • e.g. so would be the factor

  • Make the factor equal to zero and solve to find the root

    • e.g. so would be a root

How do I use the factor theorem to find unknown values?

• You might be given a polynomial equation with one or two unknown coefficients

  • e.g.

• You will be given one or two factors of the polynomial expression

  • e.g. is a factor

• To find the value(s) of the unknown(s)

  • Set the factor equal to zero and solve for

    • e.g.

  • Substitute the value of into the polynomial equation and simplify

    • e.g. simplifies to

  • Solve the resulting equation(s) to find the unknown value(s)

    • e.g.

How do I use the factor theorem to fully factorise a polynomial?

• Find a root of the polynomial equation

  • Test values of the form into the polynomial where

    • is a factor of polynomial's constant term

    • is a factor of the coefficient of the polynomial's leading term

  • e.g. for try

    • and

• Use the factor theorem to identify factors

  • e.g. and

• Divide the polynomial by each factor

  • You can divide by one factor at a time

    • e.g.

    • e.g.

  • Or divide by the expansion of the factors

    • e.g.

• Check to see if the new polynomial can be factorised

• Repeat the steps until you get a linear expression or a polynomial that cannot be factorised

• The factorised form is the product of all the factors you found

  • e.g. try

Remainder theorem

What is the remainder theorem? 

• The remainder theorem states that for any polynomial function the following is true

  • The remainder when is divided by is equal to

• The remainder theorem is used to find the remainder when you divide a polynomial function by a linear function

  • where

    • is the quotient polynomial

    • is the remainder

  • This means

    • So

• For example:

  • 

    • so the remainder when you divide by is 105

• The factor theorem is a special case of the remainder theorem when the remainder is zero

How do I use the remainder theorem?

• A question will involve a polynomial being divided by a linear expression

• You will be asked to find one of the following:

  • an unknown coefficient of the polynomial

  • an unknown coefficient of the linear expression

  • the remainder

• Follow these steps for all the cases:

  • Set the linear expression equal to zero and solve for

    • e.g.

  • Substitute the value of into the polynomial equation

    • e.g. simplifies to

  • This is equal to the remainder

    • 

  • Solve to find any unknowns

    •