Question

CAN ANYONE PLEASE SOLVE ALL LEVEL ONE QUESTION PLEASE,
I KINDLY REQUEST YOU
I REALLY NEED HELP

Answer 1

The answers are in order.

Level - 1

Obtain all zeros

The polynomial will be a multiple of $(x+\sqrt{\frac{3}{2} } )(x-\sqrt{\frac{3}{2} } )$, factor theorem

Expand the multiplication, $x^2-\frac{3}{2}$

Divide by it, but remember,

Two other roots come from the quotient.

Quotient : $2x^2-2x-4=2*(x+1)(x-2)$

$2*(x+1)(x-2)=0$ ∴$x=-1,2$

Therefore, two other roots will be -1 and 2

Find the value of k

Relation between roots and coefficients, these two

$\alpha +\beta =k+6$

$\alpha \times \beta =2(2k-1)$

From the question, given this one

$k+6=2k-1$ ∴$k=7$

Therefore, the value is 7

Frame a quadratic polynomial

Two roots are : $2+\sqrt{5}$ and $2-\sqrt{5}$

Their sum = 4, Their multiplication = -1

Now, recall relation between roots and coefficient.

The polynomial will be

x² - (sum of roots)x + (multiplication of roots)

Therefore, the polynomial is $x^2-4x-1$.

Obtain all zeros

The polynomial $x^4-7x^2+12$

can be factorized into $(x^2-3)(x^2-4)$.

The two roots come from left bracket, $x^2-3$.

Then, the other roots will come from $x^2-4$.

Therefore, all zeros are $\sqrt{3}$, $-\sqrt{3}$, $2$, $-2$

The quadratic polynomial, ...

x² - (sum of roots)x + (multiplication of roots)

Therefore, the polynomial is (c) $x^2-2x-8$

I hope you understand and it helped.