C1 Edexcel - a few questions im struggling on (+ve rep up for grabs)

The cubic polynomial 2x^3 + x^2 + kx - 10 has a factor x-2. Find the value of k

the questions i've been doing in class usually tell u the quadratic factor too in terms of a, b and c. any ideas on how to do this without the quadratic factor given to u?

The cubic polynomial 2x^3 + x^2 + kx - 10 has a factor x-2. Find the value of k

the questions i've been doing in class usually tell u the quadratic factor too in terms of a, b and c. any ideas on how to do this without the quadratic factor given to u?


Given that f(x) = 2x^3 - x^2 - 10x + 8, find a linear factor and express f(x) as the product of a linear factor and a quadratic factor.

kinda similar, but this time, how do i find the LINEAR factor to begin with?


Prove that, for all real values of k, the roots of x^2 + 2kx - 7 = 0 are real and different.

my working: b^2 - 4ac > 0 => 4k^2 + 28 > 0
however, if i equate 4k^2 + 28 to 0, i'll end up with a negative square root (complex number). where have i gone wrong? or whats the next stage


ALL help is highly appreciated! +ve rep for most helpful answer!

For the last one, you have to prove that $b^2-4ac>0$ you can't start off with the assertion that it is true. Why is $4k^2+28 > 0$ for all real k?

For the last question, I don't really think you need to equate it to zero; you've found the discriminant to be 4k^2+28 - any number, when squared, will give a positive number, then multiplying by four and adding 28 will keep it positive, so the discriminant will be always be greater than zero, so the equation will always have two real and distinct solutions

how else would u prove that b^2-4ac>0 without substituting in the values?

Read what "Revolution is my Name" said. Any real number squared is positive (or zero). So 4k^2 is positive or zero, therefore 4k^2 + 28 >0

The cubic polynomial 2x^3 + x^2 + kx - 10 has a factor x-2. Find the value of k



Given that f(x) = 2x^3 - x^2 - 10x + 8, find a linear factor and express f(x) as the product of a linear factor and a quadratic factor.



Prove that, for all real values of k, the roots of x^2 + 2kx - 7 = 0 are real and different.

my working: b^2 - 4ac > 0 => 4k^2 + 28 > 0
however, if i equate 4k^2 + 28 to 0, i'll end up with a negative square root (complex number). where have i gone wrong? or whats the next stage


ALL help is highly appreciated! +ve rep for most helpful answer!