Divide polynomials and state quotient and remainder lesson | Mathematical Methods (Year 12, VCE Units 3 & 4, Solving equations, Polynomial equations and factorisation)

Polynomial division

Align descending powers, divide leading terms, subtract the complete divisor multiple and repeat until the remainder degree is lower than the divisor

D(x)=V(x)Q(x)+R(x),\qquad\deg R<\deg V

Keep zero columns Write a zero coefficient for every missing power before aligning the division \(\displaystyle D(x)=x^3+0x^2-4x+8\)
Divide leading terms Divide the leading term of the current dividend by the leading term of the divisor \(\displaystyle \frac{x^3}{x}=x^2\)
Subtract the full multiple Multiply the whole divisor by the new quotient term, then subtract every term \(\displaystyle x^3+0x^2-4x+8-x^2\left(x-2\right)=2x^2-4x+8\)
Check the identity The remainder must have lower degree than the divisor and satisfy the division identity \(\displaystyle D(x)=V(x)Q(x)+R(x)\)

Divide the displayed polynomials and state the quotient and remainder

\frac{2x^3+0x^2-5x+6}{x-1}

\(\displaystyle D(x)=2x^3+0x^2-5x+6,\quad V(x)=x-1\)Write the zero coefficient for the missing squared term so all powers remain aligned
\(\displaystyle \frac{2x^3}{x}=2x^2,\quad D(x)-2x^2V(x)=2x^2-5x+6\)The displayed division gives the first quotient term; subtract its full divisor multiple to find the next dividend
\(\displaystyle \frac{2x^2}{x}=2x,\quad (2x^2-5x+6)-2xV(x)=-3x+6\)The displayed division gives the next quotient term; subtract its full divisor multiple to find the next dividend
\(\displaystyle \frac{-3x}{x}=-3,\quad (-3x+6)-(-3)V(x)=3,\quad Q(x)=2x^2+2x-3,\quad R(x)=3\)The remainder is 3, whose degree is lower than the linear divisor

Dropping a term during subtraction

✕\(\displaystyle x^3-4x+8-x^2\left(x-2\right)=-4x+8\)✓\(\displaystyle x^3-4x+8-x^2\left(x-2\right)=2x^2-4x+8\)

Subtracting x^2(x-2) adds 2x^2 to the result

Adding instead of subtracting

✕\(\displaystyle 2x^3+0x^2-5x+6+2x^2\left(x-1\right)=4x^3-2x^2-5x+6\)✓\(\displaystyle 2x^3+0x^2-5x+6-2x^2\left(x-1\right)=2x^2-5x+6\)

Each divisor multiple must be subtracted from the current dividend

Leaving a high-degree remainder

✕\(\displaystyle D(x)=(x-1)(2x^2+2x-3)+(3x+1)\)✓\(\displaystyle D(x)=(x-1)(2x^2+2x-3)+3\)

The remainder must have lower degree than the divisor, which is linear here

introIdentify quotient and remainder for a simple polynomial division

standardDivide a polynomial with a missing power and state the quotient and remainder

challengeDivide by a non-monic linear or quadratic divisor and check the identity

examComplete a full polynomial division and use the remainder or identity in a follow-up condition

Why must a missing power be written with coefficient zero?+

The zero coefficient preserves the position of every other power. This keeps the subtraction columns aligned

When does polynomial long division stop?+

Stop when the current remainder has lower degree than the divisor. It can then be written as the remainder

How can a polynomial division answer be checked?+

Substitute the quotient and remainder into the division identity displayed above, and confirm that it expands to the original dividend

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 1: Solving equations