Quartic Polynomial Factorisation: 5-i is root of 2z^3+az^2+62z+(a-5)

[Onigma]

Hi,

I was presented with the following question (question 4).



I was able to complete it using the below method.... Notice how in the bottom line, in the second lot of bracket, the z term has a coefficient of 2.



I was then presented with the following similar question (question 7).



I was able to begin factorizing it using the below method. Notice how there is no coefficient of x in the second lot of brackets like last time. Now it is out the front of the entire equation.


I understand the process of equating coefficents after this point to find the values of the unknowns. What I don't understand is in the first example why the coefficient of x can come into the second lot of brackets, while in the second example, the coefficient of x (2) has to stay out the front. When I attempted to use the alternate method in either example, it completely ruins it and gives a wrong answer. Can anyone please provide some explanation as to how/why/when you use either method or what I'm missing/forgetting/doing wrong?

Thanks

 

[lev888]

It doesn't matter where the coefficient is as long as you solve for x correctly.
(2x-2)(blah)=0, 2x-2=0, x=1.
2(x-2)(blah)=0, x-2=0, x=2.

 

[Onigma]

It doesn't matter where the coefficient is as long as you solve for x correctly.
(2x-2)(blah)=0, 2x-2=0, x=1.
2(x-2)(blah)=0, x-2=0, x=2.

Doesn't by having the coefficent of 2 out the front of the whole polynomial though, then effect the x values from the other factor?

 

[lev888]

Doesn't by having the coefficent of 2 out the front of the whole polynomial though, then effect the x values from the other factor?

Sure, different polynomials have different factors and zeros.
2x-2 and 2(x-2) are different polynomials.

2x-2 and 2(x-1) are the same.

Please let me know if I misunderstood your question.

 

[JeffM]

Hi,

I was presented with the following question (question 4).

View attachment 9223

I was able to complete it using the below method.... Notice how in the bottom line, in the second lot of bracket, the z term has a coefficient of 2.

View attachment 9224

I was then presented with the following similar question (question 7).

View attachment 9225

I was able to begin factorizing it using the below method. Notice how there is no coefficient of x in the second lot of brackets like last time. Now it is out the front of the entire equation.
View attachment 9226

I understand the process of equating coefficents after this point to find the values of the unknowns. What I don't understand is in the first example why the coefficient of x can come into the second lot of brackets, while in the second example, the coefficient of x (2) has to stay out the front. When I attempted to use the alternate method in either example, it completely ruins it and gives a wrong answer. Can anyone please provide some explanation as to how/why/when you use either method or what I'm missing/forgetting/doing wrong?

Thanks

Why are you worrying about problem 7 when you have not completed problem 4.

You were to find the both the other zeroes of P(z) and the value of a, given that:

\(\displaystyle 5 - i \text { is a zero of } P(z) = 2z^3 + az^2 + 62z + a - 5 \text { and } a \in \mathbb R.\)

You remembered that, because P(z) has only real coefficients, any complex roots come in conjugate pairs. So you correctly identified
the other complex zero as 5 + i, which gave you one third of the answer. But because P(z) has only real coefficients and is of odd degree, it must have at least one real zero, making the third zero. You forgot that, and you forgot to find a.

You started well by realizing that

\(\displaystyle \{z - (5 - i)\}\{z - (5 + i)\} = z^2 - z(5 + i + 5 - i) + (5 - i)(5 + i) = z^2 - 10z + 26\)

factors P(z) so \(\displaystyle P(z) = (z^2 - 10z + 26)(2z + b).\)

But that does not finish up the two missing parts of the problem.

\(\displaystyle P(z) = (z^2 - 10z + 26)(2z + b) = \)

\(\displaystyle 2z^3 + (b - 20)z^2 + (52 - 10b)z + 26b \implies\)

\(\displaystyle 62 = 52 - 10b \implies b = -\ 1 \text{, BUT}\)

\(\displaystyle 26b = a - 5 \implies a = 26b + 5 = -\ 26 + 5 = -\ 21.\)

So \(\displaystyle a = - 21.\) That is the second part of your answer.

How about the real zero?

\(\displaystyle (2z + b) = 0 \implies 2z - 1 = 0 \implies z = \dfrac{1}{2}.\)

Thus the third zero = \(\displaystyle \dfrac{1}{2}\).

Let's check.

\(\displaystyle 2 * \left ( \dfrac{1}{2} \right )^3 - 21 * \left ( \dfrac{1}{2} \right )^2 + 62 * \left ( \dfrac{1}{2} \right ) - 21 - 5 =\)

\(\displaystyle 2 * \dfrac{1}{8} - 21 * \dfrac{1}{4} + 62 * \dfrac{1}{2} - 26 =\)

\(\displaystyle \dfrac{1}{4} - \dfrac{21}{4} + 31 - 26 = -\ \dfrac{20}{4} + 5 = 0.\)

I am not at all sure what you are asking at the end. I think you are talking about treatment of the leading coefficient (the coefficient of the cubed term). If so, that is a good question. Please confirm