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Algebraic divisionEdexcel International A Level Maths: Revision notes

Section 1

Dividing polynomials

When a polynomial f(x)f(x) is divided by a linear expression (ax+b)(ax+b), the result can be written f(x)=(ax+b) Q(x)+R.f(x)=(ax+b)\,Q(x)+R. Here (ax+b)(ax+b) is the divisor, Q(x)Q(x) is the quotient and RR is the remainder, a constant. If f(x)f(x) has degree 33, the quotient has degree 22. The remainder has lower degree than the divisor, so it is a number.

Only division by (ax+b)(ax+b) or (ax−b)(ax-b) is needed. If the remainder is 00 the division is exact, and (ax+b)(ax+b) is a factor of f(x)f(x).

Key termsdivisorquotientremainder
Common mistake

Writing the remainder as part of the quotient, such as x2+4x+3+7x−2x^2+4x+3+\frac{7}{x-2}. The quotient is x2+4x+3x^2+4x+3 and the remainder is 77.

Section 2

Long division by (x−a)(x-a)

Divide f(x)=x3+2x2−5x+1f(x)=x^3+2x^2-5x+1 by (x−2)(x-2).

  1. Divide the leading terms: x3÷x=x2x^3\div x=x^2. Multiply back: x2(x−2)=x3−2x2x^2(x-2)=x^3-2x^2. Subtract from f(x)f(x): 4x2−5x4x^2-5x.
  2. 4x2÷x=4x4x^2\div x=4x. 4x(x−2)=4x2−8x4x(x-2)=4x^2-8x. Subtract: 3x+13x+1.
  3. 3x÷x=33x\div x=3. 3(x−2)=3x−63(x-2)=3x-6. Subtract: 77.

The quotient is x2+4x+3x^2+4x+3 and the remainder is 77, so x3+2x2−5x+1=(x−2)(x2+4x+3)+7.x^3+2x^2-5x+1=(x-2)(x^2+4x+3)+7.

Key termslong division
Exam tip

If a power is missing, insert it with coefficient 00. To divide x3+5x−4x^3+5x-4, write x3+0x2+5x−4x^3+0x^2+5x-4.

Section 3

Dividing by (ax+b)(ax+b)

The method is the same, but dividing the leading terms involves the coefficient aa. Divide g(x)=6x3+x2−5x+7g(x)=6x^3+x^2-5x+7 by (2x+1)(2x+1).

  1. 6x3÷2x=3x26x^3\div2x=3x^2. 3x2(2x+1)=6x3+3x23x^2(2x+1)=6x^3+3x^2. Subtract: −2x2−5x-2x^2-5x.
  2. −2x2÷2x=−x-2x^2\div2x=-x. −x(2x+1)=−2x2−x-x(2x+1)=-2x^2-x. Subtract: −4x+7-4x+7.
  3. −4x÷2x=−2-4x\div2x=-2. −2(2x+1)=−4x−2-2(2x+1)=-4x-2. Subtract: 99.

The quotient is 3x2−x−23x^2-x-2 and the remainder is 99.

Check by expanding: (2x+1)(3x2−x−2)+9=6x3+x2−5x−2+9=g(x)(2x+1)(3x^2-x-2)+9=6x^3+x^2-5x-2+9=g(x).

Key termsleading term
Common mistake

Subtracting carelessly when a term is negative. Write the subtraction out line by line and watch the signs.

Section 4

The remainder without dividing

The remainder when f(x)f(x) is divided by (ax+b)(ax+b) is f(−ba)f\left(-\frac{b}{a}\right). This is the remainder theorem. It follows from f(x)=(ax+b)Q(x)+Rf(x)=(ax+b)Q(x)+R, because substituting x=−bax=-\frac{b}{a} makes the bracket zero.

Examples:

  • Divide by (x−2)(x-2): the remainder is f(2)f(2). For f(x)=x3+2x2−5x+1f(x)=x^3+2x^2-5x+1 this is 8+8−10+1=78+8-10+1=7.
  • Divide by (2x+1)(2x+1): the remainder is g(−12)g\left(-\frac12\right). For g(x)=6x3+x2−5x+7g(x)=6x^3+x^2-5x+7 this is −34+14+52+7=9-\frac34+\frac14+\frac52+7=9.
  • Divide by (x+2)(x+2): the remainder is g(−2)=−48+4+10+7=−27g(-2)=-48+4+10+7=-27.

Use this to check a long division, or when only the remainder is wanted.

Key termsremainder theorem
Common mistake

Substituting the wrong sign. For (x+2)(x+2) use x=−2x=-2, and for (2x−1)(2x-1) use x=12x=\frac12.

Section 5

Finding unknown constants

Remainder information gives equations in unknown coefficients. Let f(x)=3x3+ax2−7x+bf(x)=3x^3+ax^2-7x+b leave remainder 1212 on division by (x−2)(x-2) and −23-23 on division by (3x+1)(3x+1).

f(2)=12f(2)=12: 24+4a−14+b=1224+4a-14+b=12, so 4a+b=24a+b=2. f(−13)=−23f\left(-\frac13\right)=-23: −19+a9+73+b=−23-\frac19+\frac{a}{9}+\frac73+b=-23. Multiply by 99: a+9b=−227a+9b=-227.

Solve simultaneously: a=7a=7 and b=−26b=-26.

When a question gives two remainders, write one equation from each and solve them as simultaneous equations. Clear fractions before solving.

Key termssimultaneous equations
Exam tip

Always check your constants by substituting them back into one of the remainder equations.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Algebraic division

  1. The polynomial f(x)=x3+2x2−5x+1f(x)=x^3+2x^2-5x+1 is divided by (x−2)(x-2).
    Use the remainder theorem to check your remainder.2 marks
  2. Let g(x)=6x3+x2−5x+7g(x)=6x^3+x^2-5x+7.
    Find the remainder when g(x)g(x) is divided by (x+2)(x+2).2 marks
  3. Let f(x)=4x3−8x2+x+6f(x)=4x^3-8x^2+x+6.
    Find the quotient and the remainder when f(x)f(x) is divided by (2x−1)(2x-1).3 marks
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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).