Polynomial Division (DP IB Analysis & Approaches (AA): HL): Revision Note

Polynomial division

What is polynomial division?

• Polynomial division is the process of dividing two polynomials

  • $\frac{P\left(x\right)}{D\left(x\right)}$

    • $D\left(x\right)$ is called the divisor

    • The degree of the divisor is less than or equal to the degree of the polynomial

• The result gives a quotient polynomial $Q\left(x\right)$ and a remainder polynomial $R\left(x\right)$

  • $\frac{P\left(x\right)}{D\left(x\right)}=Q\left(x\right)+\frac{R\left(x\right)}{D\left(x\right)}$

What are the degrees of the quotient and remainder?

• If $P\left(x\right)$ has degree $n$ and is divided by a divisor $D\left(x\right)$ with degree $k\le n$

  •  $\frac{P\left(x\right)}{D\left(x\right)}=Q\left(x\right)+\frac{R\left(x\right)}{D\left(x\right)}$

• The degree of the quotient $Q\left(x\right)$ is equal to $n-k$

• The degree of the remainder $R\left(x\right)$ is less than $k$

• For example, when $x^5-7$ is divided by $x^2+2$

  • The degree of the quotient is 3

  • The degree of the remainder is less than 2

    • It could be 0 (a constant term) or 1 (a linear expression)

How do I divide polynomials?

• Let's use the example:

  • $P\left(x\right)=2x^4+3x^3-x^2+5$

  • $D\left(x\right)=x^2+2x-1$

• STEP 1
  Divide the leading term of the polynomial $P\left(x\right)$ by the leading term of the divisor

  • This is the first term of the quotient

  • e.g. $\frac{2x^4}{x^2}=2x^2$

• STEP 2
  Multiply the divisor by this term

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

  Subtract this from the original polynomial $P\left(x\right)$ to find the current remainder

  • The leading term should be cancelled out

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

• STEP 3
  Repeat steps 1 – 2 using the current remainder as the main polynomial

  • Keep repeating the steps until the degree of the remainder is less than the degree of the division

  • Find the second term of the quotient

    • e.g. $-\frac{x^3}{x^2}=-x$

    • e.g. $-x\left(x^2+2x-1\right)=-x^3-2x^2+x$

    • e.g. $\left(-x^3+x^2+5\right)-\left(-x^3-2x^2+x\right)=3x^2-x+5$

  • Find the third term of the quotient

    • e.g. $\frac{3x^2}{x^2}=3$

    • e.g. $3\left(x^2+2x-1\right)=3x^2+6x-3$

    • e.g. $\left(3x^2-x+5\right)-\left(3x^2+6x-3\right)=-7x+8$

• STEP 5
  Identify the quotient and the remainder

  • The quotient is the sum of all the terms from step 1

    • e.g. $Q\left(x\right)=2x^2-x+3$

  • The remainder is the last remainder from step 2

    • e.g. $R\left(x\right)=-7x+8$

How do I divide by comparing coefficients?

• STEP 1
  Write the expression as $P\left(x\right)=Q\left(x\right)D\left(x\right)+R\left(x\right)$

  • Use the facts about the degrees to get the correct number of terms

    • e.g. $2x^4+3x^3-x^2+5=\left(x^2+2x-1\right)\left(a{x}^2+bx+c\right)+\left(dx+f\right)$

• STEP 2
  Work out the leading coefficient of the polynomial on the right-hand side and set it equal to the leading coefficient on the left-hand side

  • You can find the leading term of the quotient

    • e.g. for $x^4$: $2=a$ therefore $a=2$

• STEP 3
  Repeat the step for the next leading term

  • You might have to use the previous value

    • e.g. for $x^3$: $3=b+2a=b+4$ therefore $b=-1$

• STEP 4
  Keep repeating to find all the unknowns

  • Remember to include missing terms such as $0x$

    • e.g. for $x^2$: $-1=c+2b-a=c-2-2=c-4$ therefore $c=3$

    • e.g. for $x$: $0=2c-b+d=6+1+d=7+d$ therefore $d=-7$

    • e.g. for constant terms: $5=-c+f=-3+f$ therefore $f=8$