Factor & Remainder Theorem (DP IB Maths: AA HL)

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Lucy

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Lucy

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Factor Theorem

What is the factor theorem?

  • The factor theorem is used to find the linear factors of polynomial equations
  • This topic is closely tied to finding the zeros and roots of a polynomial function/equation
    • As a rule of thumb a zero refers to the polynomial function and a root refers to a polynomial equation
  • For any polynomial function P(x)
    • (x - k) is a factor of P(x) if P(k) = 0
    • P(k) = 0 if (x - k) is a factor of P(x)

How do I use the factor theorem?

  • Consider the polynomial function P(x) = anxn + an-1xn-1 + … + a1xa0 and (x - k) is a factor
    • Then, due to the factor theorem P(k) = ankn + an-1kn-1 + … + a1k + a0 = 0
    • P left parenthesis x right parenthesis equals left parenthesis x minus k right parenthesis cross times Q left parenthesis x right parenthesis, where Q(x) is a polynomial that is a factor of P(x)
    • Hence, fraction numerator P left parenthesis x right parenthesis over denominator x minus k end fraction equals Q left parenthesis x right parenthesis , where Q(x) is another factor of P(x)
  • If the linear factor has a coefficient of x then you must first factorise out the coefficient
    • If the linear factor is left parenthesis a x blank – blank b right parenthesis blank equals a open parentheses x minus b over a close parentheses rightwards arrow P open parentheses b over a close parentheses equals 0

Examiner Tip

  • A common mistake in exams is using the incorrect sign for either the root or the factor
  • If you are asked to find integer solutions to a polynomial then you only need to consider factors of the constant term

Worked example

Determine whether left parenthesis x minus 2 right parenthesis is a factor of the following polynomials:

a)
space f left parenthesis x right parenthesis equals x cubed minus 2 x squared minus x plus 2.

page1

b)
space g left parenthesis x right parenthesis equals 2 x cubed plus 3 x squared minus x plus 5.

2-7-1-ib-aa-hl-factor-theorem-b-we-solution

It is given that left parenthesis 2 x minus 3 right parenthesis is a factor of space h left parenthesis x right parenthesis equals 2 x cubed minus b x squared plus 7 x minus 6.

c)
Find the value of b.

mZEjMdDm_2-7-1-ib-aa-hl-factor-theorem-c-we-solution

Remainder Theorem

What is the remainder theorem? 

  • The remainder theorem is used to find the remainder when we divide a polynomial function by a linear function
  • When any polynomial P(x) is divided by any linear function (x - k) the value of the remainder R is given by P(k) = R
    • Note, when P(k) = 0 then (x - k) is a factor of P(x)

How do I use the remainder theorem?

  • Consider the polynomial function P(x) = anxn + an-1xn-1 + … + a1xa0 and the linear function (x - k
    • Then, due to the remainder theorem P(k) = ankn + an-1kn-1 + … + a1k + a0 = R
    • P left parenthesis x right parenthesis equals left parenthesis x minus k right parenthesis cross times Q left parenthesis x right parenthesis plus R, where Q(x) is a polynomial
    • Hence, fraction numerator P left parenthesis x right parenthesis over denominator x minus k end fraction equals Q left parenthesis x right parenthesis plus fraction numerator R over denominator x minus k end fraction , where R is the remainder
  • If the linear function has a coefficient of x then you must first factorise out the coefficient
    • If the linear function is left parenthesis a x blank – blank b right parenthesis blank equals a open parentheses x minus b over a close parentheses rightwards arrow P open parentheses b over a close parentheses equals R

Worked example

Let space f left parenthesis x right parenthesis equals 2 x to the power of 4 minus 2 x cubed minus x squared minus 3 x plus 1, find the remainder R when space f left parenthesis x right parenthesis is divided by:

a)
x minus 3.

2-7-1-ib-aa-hl-remainder-theorem-a-we-solution

b)
x plus 2.

2-7-1-ib-aa-hl-remainder-theorem-b-we-solutionThe remainder when space f left parenthesis x right parenthesis is divided by left parenthesis 2 x plus k right parenthesis is 893 over 8.

c)
Given that k greater than 0, find the value of k.

2-7-1-ib-aa-hl-remainder-theorem-c-we-solution

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Lucy

Author: Lucy

Expertise: Head of STEM

Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels. Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all. Lucy has created revision content for a variety of domestic and international Exam Boards including Edexcel, AQA, OCR, CIE and IB.