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

Factor theorem

What is the factor theorem?

• The factor theorem states that for any polynomial function $P\left(x\right)$ the following is true

  • If $\left(ax-b\right)$ is a factor of $P\left(x\right)$ then $P\left(\frac{b}{a}\right)=0$

  • If $P\left(\frac{b}{a}\right)=0$ then $\left(ax-b\right)$ is a factor of $P\left(x\right)$

• For example:

  • $2{\left(-3\right)}^4+5{\left(-3\right)}^3-27=0$

    • so $\left(x+3\right)$ is a factor of $2x^4+5x^3-27=0$

  • $2x^4-x^3+4x^2-8x+3\equiv \left(2x-1\right)\left(x^3+2x-3\right)$

    • so $2{\left(\frac{1}{2}\right)}^4-{\left(\frac{1}{2}\right)}^3+4{\left(\frac{1}{2}\right)}^2-8\left(\frac{1}{2}\right)+3=0$

• 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. $x=\frac{1}{2}\Rightarrow 2x-1=0$ so $\left(2x-1\right)$ would be the factor

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

    • e.g. $3x+4=0\Rightarrow x=-\frac{4}{3}$ so $x=-\frac{4}{3}$ 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. $2x^3-x^2+ax-3=0$

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

  • e.g. $\left(2x-3\right)$ is a factor

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

  • Set the factor equal to zero and solve for $x$

    • e.g. $2x-3=0\Rightarrow x=\frac{3}{2}$

  • Substitute the value of $x$ into the polynomial equation and simplify

    • e.g. $2{\left(\frac{3}{2}\right)}^3-{\left(\frac{3}{2}\right)}^2+a\left(\frac{3}{2}\right)-3=0$ simplifies to $\frac{3}{2}a+\frac{3}{2}=0$

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

    • e.g. $a=-1$

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

• Find a root of the polynomial equation

  • Test values of the form $x=\pm \frac{p}{q}$ into the polynomial where

    • $p$ is a factor of polynomial's constant term

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

  • e.g. for $P\left(x\right)=2x^4+5x^3-x^2+5x-3$ try $x=\pm 1,\pm 3,\pm \frac{1}{2},\pm \frac{3}{2}$

    • $P\left(\frac{1}{2}\right)=0$ and $P\left(-3\right)=0$

• Use the factor theorem to identify factors

  • e.g. $\left(2x-1\right)$ and $\left(x-3\right)$

• Divide the polynomial by each factor

  • You can divide by one factor at a time

    • e.g. $\frac{2x^4+5x^3-x^2+5x-3}{2x-1}=x^3+3x^2+x+3$

    • e.g. $\frac{x^3+3x^2+x+3}{x+3}=x^2+1$

  • Or divide by the expansion of the factors

    • e.g. $\frac{2x^4+5x^3-x^2+5x-3}{2x^2+5x-3}=x^2+1$

• 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. $2x^4+5x^3-x^2+5x-3=\left(2x-1\right)\left(x+3\right)\left(x^2+1\right)$ try

Remainder theorem

What is the remainder theorem? 

• The remainder theorem states that for any polynomial function $P\left(x\right)$ the following is true

  • The remainder when $P\left(x\right)$ is divided by $\left(ax-b\right)$ is equal to $P\left(\frac{b}{a}\right)$

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

  • $\frac{P\left(x\right)}{ax-b}=Q\left(x\right)+\frac{r}{ax-b}$ where

    • $Q\left(x\right)$ is the quotient polynomial

    • $r$ is the remainder

  • This means $P\left(x\right)=\left(ax-b\right)Q\left(x\right)+r$

    • So $P\left(\frac{b}{a}\right)=0\times Q\left(\frac{b}{a}\right)+r=r$

• For example:

  • $8{\left(\frac{5}{2}\right)}^3-2\left(\frac{5}{2}\right)-15=105$

    • so the remainder when you divide $\left(8x^3-2x-15\right)$ by $\left(2x-5\right)$ 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 $x$

    • e.g. $3x+2=0\Rightarrow x=-\frac{2}{3}$

  • Substitute the value of $x$ into the polynomial equation

    • e.g. $9{\left(-\frac{2}{3}\right)}^3+a\left(-\frac{2}{3}\right)+5$ simplifies to $\frac{7}{3}-\frac{2}{3}a$

  • This is equal to the remainder

    • $\frac{7}{3}-\frac{2}{3}a=1$

  • Solve to find any unknowns

    • $a=2$