MYTUTOR SUBJECT ANSWERS

608 views

Let P(z) = z⁴ + az³ + bz² + cz + d be a quartic polynomial with real coefficients. Let two of the roots of P(z) = 0 be 2 – i and -1 + 2i. Find a, b, c and d.

This is a question from a FP1 paper. Here i denotes √-1.

Fact that should be burned into your soul: ‘Complex roots of a polynomial equation with real coefficients form conjugate pairs’. This tells you if z = x + yi is a root then so is its conjugate z* = x – yi. Using this fact, we can deduce that 2 + i and -1 – 2i are also roots of P(z) = 0.

So we now have 4 roots of our quartic equation P(z) = 0, so that’s all of them! We can now employ the factor theorem that you (probably) met in C1. Remember, this states that if z = α is a root of a polynomial then (z – α) is a factor of that polynomial.

So, since we’re told the leading coefficient of P(z) is 1 we can apply the factor theorem to deduce that

P(z) = (z – (2 - i))(z – (2 + i))(z – (-1 + 2i))(z – (-1 - 2i))

So now all it comes down to is some tedious expansion of brackets and a bit of simplification. It’s a good exercise to build your confidence with complex numbers.

⇒ P(z) = (z – 2 + i)(z – 2 – i)(z + 1 – 2i)(z + 1 + 2i)

⇒ P(z) = (z² - 2z – zi – 2z + 4 + 2i + zi - 2i - i²)(z² + z + 2zi + z + 1 + 2i – 2zi – 2i - 4i²)

⇒ P(z) = (z² - 4z + 5)(z² + 2z + 5)                           [Remember: i² = - 1 by definition]

⇒ P(z) = z⁴ + 2z³ + 5z² - 4z³ - 8z² - 20z  + 5z² + 10z + 25

⇒ P(z) = z⁴ - 2z³ + 2z² - 10z + 25

Thus a = -2, b = 2, c = -10, d = 25 and we’re done.

George B. GCSE Maths tutor, A Level Maths tutor, A Level Further Math...

2 years ago

Answered by George, an A Level Further Mathematics tutor with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

89 SUBJECT SPECIALISTS

Katrina M. IB Maths tutor, 13 Plus  Maths tutor, GCSE Maths tutor, A ...
£20 /hr

Katrina M.

Degree: Mathematics Bsc (Bachelors) - Exeter University

Subjects offered:Further Mathematics , Physics+ 2 more

Further Mathematics
Physics
Maths
-Personal Statements-

“Hi, I am a patient, experienced and enthusiastic tutor ready to help each student reach their full potential.”

£20 /hr

Jacob F.

Degree: Mathematics (Bachelors) - Warwick University

Subjects offered:Further Mathematics , Maths+ 2 more

Further Mathematics
Maths
German
-Personal Statements-

“2nd year Maths undergrad with enhanced DBS (CRB) certification and experience teaching in secondary schools and at University. Fluent German speaker.”

Roma V. A Level Maths tutor, 13 Plus  Maths tutor, GCSE Maths tutor, ...
£26 /hr

Roma V.

Degree: Mathematics, Operational Research, Statistics and Economics (Bachelors) - Warwick University

Subjects offered:Further Mathematics , Maths+ 1 more

Further Mathematics
Maths
Economics

“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

About the author

PremiumGeorge B. GCSE Maths tutor, A Level Maths tutor, A Level Further Math...
£26 /hr

George B.

Degree: Mathematics (Masters) - Warwick University

Subjects offered:Further Mathematics , Maths+ 1 more

Further Mathematics
Maths
.STEP.

“Fourth year undergraduate at one of the top universities for Maths. Eager to tutor and help improve your grades.”

You may also like...

Posts by George

Given that a and b are distinct positive numbers, find a polynomial P(x) such that the derivative of f(x) = P(x)e^(−x²) is zero for x = 0, x = ±a and x = ±b, but for no other values of x.

How do I solve Hannah’s sweet question?

Let P(z) = z⁴ + az³ + bz² + cz + d be a quartic polynomial with real coefficients. Let two of the roots of P(z) = 0 be 2 – i and -1 + 2i. Find a, b, c and d.

What values of θ between 0 and 2π satisfy the equation cosec(θ) + 5cot(θ) = 3sin(θ)?

Other A Level Further Mathematics questions

Integrate (x+4)/(x^2+2x+2)

How to approximate the Binomial distribution to the Normal Distribution

Given that f(x)=2sinhx+3coshx, solve the equation f(x)=5 giving your answers exactly.

Solve the inequality x^3 + x^2 > 6x

View A Level Further Mathematics tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok