FP 1 Complex numbers

Given that 2 + 3i is the root of the equation z^4 +40 z + k =0 find the value if k and write down another root of the equation

I got k = 39 and then using the factor theorem found -1 and -3 BUT the other factor is 2 - 3i which I do not know how to get.

I would like a pointer please.

Given that 2 + 3i is the root of the equation z^4 +40 z + k =0 find the value if k and write down another root of the equation

I got k = 39 and then using the factor theorem found -1 and -3 BUT the other factor is 2 - 3i which I do not know how to get.

I would like a pointer please.

Whenever a complex number is the root of a polynomial, its complex conjugate will also be a root. This means that if a+bi is a root, a-bi will also be a root.

In this case, since 2+3i is a root, you can immediately say without any working out that 2-3i is also a root because of the complex conjugate rule.

Whenever a complex number is the root of a polynomial, its complex conjugate will also be a root. This means that if a+bi is a root, a-bi will also be a root. In this case, since 2+3i is a root, you can immediately say without any working out that 2-3i is also a root because of the complex conjugate rule.

Have another look through your textbook

You should have a statement which tells you that if a polynomial equation has real coefficients then any complex roots occur in conjugate pairs, i.e. if a + ib is a root then so is a - ib.

This is only true when the polynomial has real coefficients.

It's fairly clear that it can't be true in general, since we can always form a polynomial such as:

$f(z)=(z-i)(z-1) = z^2-(1+i)z+i$

whose roots are *not* complex conjugates.



because the polynomial in question has real coefficients.

This is only true when the polynomial has real coefficients.

It's fairly clear that it can't be true in general, since we can always form a polynomial such as:

$f(z)=(z-i)(z-1) = z^2-(1+i)z+i$

whose roots are *not* complex conjugates.

because the polynomial in question has real coefficients.

From the title, you can see that this question is about FP1. What you're saying is absolutely correct but all polynomial equations in FP1 have real coefficients so the OP doesn't really need to worry about this

I disagree, for a couple of reasons.

1. As it stands, the claim that you made earlier is simply incorrect, and liable to mislead anyone reading it at some later time.

This is because you didn't limit its scope to polynomial equations used in FP1, except rather weakly by context. Even if you had done so, that's not particularly useful since it puts the onus on the reader to find out the nature of polynomial equations used in FP1, even assuming that they happen to notice the context.

Someone chancing across your statement from the USA, for example, isn't in a good position to do so. It's always better in mathematics to make a precise and unambiguous statement.

2. The OP really does need to worry about it.


That's because it is an examinable part of the syllabus, and the examiners are quite at liberty to ask questions based on this fact. For example, I can imagine something like the following coming up:

"No complex conjugate pairs appear in the set of roots of the polynomial equation f(z) = 0, though it does have at least one complex root. Explain briefly why the coefficients of f(z) are not all real."

It would not be possible to answer this without knowledge of this fact, and a little bit of logic.

Are you sure this is in the syllabus? Could you show me where it says that? (I'm not disagreeing that we should be rigorous in our definitions!)

2. The OP really does need to worry about it.

That's because it is an examinable part of the syllabus, and the examiners are quite at liberty to ask questions based on this fact. For example, I can imagine something like the following coming up:

"No complex conjugate pairs appear in the set of roots of the polynomial equation f(z) = 0, though it does have at least one complex root. Explain briefly why the coefficients of f(z) are not all real."

It would not be possible to answer this without knowledge of this fact, and a little bit of logic.

From the title, you can see that this question is about FP1. What you're saying is absolutely correct but all polynomial equations in FP1 have real coefficients so the OP doesn't really need to worry about this

I disagree, for a couple of reasons.

1. As it stands, the claim that you made earlier is simply incorrect, and liable to mislead anyone reading it at some later time.

This is because you didn't limit its scope to polynomial equations used in FP1, except rather weakly by context. Even if you had done so, that's not particularly useful since it puts the onus on the reader to find out the nature of polynomial equations used in FP1, even assuming that they happen to notice the context.

Someone chancing across your statement from the USA, for example, isn't in a good position to do so. It's always better in mathematics to make a precise and unambiguous statement.

2. The OP really does need to worry about it.

That's because it is an examinable part of the syllabus, and the examiners are quite at liberty to ask questions based on this fact. For example, I can imagine something like the following coming up:

"No complex conjugate pairs appear in the set of roots of the polynomial equation f(z) = 0, though it does have at least one complex root. Explain briefly why the coefficients of f(z) are not all real."

It would not be possible to answer this without knowledge of this fact, and a little bit of logic.

That question would not be able to appear in FP1. I have gone through the Edexcel, AQA and OCR specifications and in none of them does it mention anything about having to know that the conjugate pair rule does not stand if the coefficients are not real. The line you quoted from the OCR specification, does not support what you're saying. The fact that it only talks about polynomial equations with real coefficients shows that you are never going to be asked a question about polynomial equations with imaginary coefficients. I accept your point that it might be of interest to other people, but it's just nonsense to claim that a polynomial with imaginary coefficients would come up in an FP1 exam.

That question would not be able to appear in FP1.

I have gone through the Edexcel, AQA and OCR specifications and in none of them does it mention anything about having to know that the conjugate pair rule does not stand if the coefficients are not real.

The line you quoted from the OCR specification, does not support what you're saying.

The fact that it only talks about polynomial equations with real coefficients shows that you are never going to be asked a question about polynomial equations with imaginary coefficients. I accept your point that it might be of interest to other people, but it's just nonsense to claim that a polynomial with imaginary coefficients would come up in an FP1 exam.

I beg to differ. The example question that I posted requires a combination of knowledge of the statement of the spec. that I quoted, plus the ability to recast it in contrapositive form. This is well within the scope of a A* question in further maths.

It's in the OCR FP1 syllabus, and presumably in the syllabus of whichever board the OP is studying from.

From the OCR syllabus, Complex Numbers section:

"(c) use the result that, for a polynomial equation with real coefficients, any non-real roots occur in conjugate pairs;"

I
"No complex conjugate pairs appear in the set of roots of the polynomial equation f(z) = 0, though it does have at least one complex root. Explain briefly why the coefficients of f(z) are not all real."

A* questions in FP1?

Based on this, I think that it would be fair if:

was asked! This question would test you on the understanding of the method you are using.

That's a pretty bold claim. It's a question that certainly is not disallowed by the wording of the spec. How can you be so certain?



That is not a good way of deciding whether or not a question is possible. There are going to be an infinite number of pieces of information that a spec does not mention.



I beg to differ. The example question that I posted requires a combination of knowledge of the statement of the spec. that I quoted, plus the ability to recast it in contrapositive form. This is well within the scope of a A* question in further maths.



Well, I disagree, since there are many ways that a question with imaginary coefficients can "come up", and I've already given an example of one above. Here's another that is consistent with the wording of the OCR spec:

"Write down the value of $a$ given that the roots of $z^3-(6+ai)z^2+13z-10=0$ are $2,2-i,2+i$"

If you feel that is inconsistent with the spec, then I'm keen to hear your reasoning. it's (ostensibly) a polynomial with complex coefficients, but the answer will be entirely obvious to anyone who understands the statement in the OCR spec, and is able to "use the result", to quote their words.