Question

Find a factorization of x² + 2x³ + 7x² - 6x + 44, given that −2+i√√7 and 1 - i√/3 are roots.

Answer 1

Surely you mean

$x^4 + 2x^3 + 7x^2 - 6x + 44$

The coefficients are integers, so the conjugate of any complex root to the quartic is also a root. In particular,

$-2 + i\sqrt7 \text{ and thus } -2 - i\sqrt7$

are roots, as are

$1 - i\sqrt3 \text{ and } 1 + i\sqrt3$

Then the complete factorization is

$\boxed{(x - (-2 + i\sqrt7)) (x - (-2 - i\sqrt7)) (x - (1 + i\sqrt3)) (x - (1 - i\sqrt3))}$

which we can simplify to get a factorization involving only integer coefficients,

$(x^2 + 4x + 11) (x^2 - 2x + 4)$

Answer 2

A factorization of $x^4+2x^3+7x^2-6x+44$ is $(x^2+4x+11)(x^2-2x+4)$.

What are the properties of roots of a polynomial?

• The maximum number of roots of a polynomial of degree $n$ is $n$.
• For a polynomial with real coefficients, the roots can be real or complex.
• The complex roots of a polynomial with real coefficients always exist in a pair of conjugate numbers i.e., if $a+ib$ is a root, then $a-ib$ is also a root.

If the roots of the polynomial $p(x)=ax^4+bx^3+cx^2+dx+e$ are $r_1,r_2,r_3,r_4$, then it can be factorized as $p(x)=(x-r_1)(x-r_2)(x-r_3)(x-r_4)$.

Here, we are to find a factorization of $p(x)=x^4+2x^3+7x^2-6x+44$. Also, given that $-2+i\sqrt{7}$ and $1-i\sqrt{3}$ are roots of the polynomial.

Since $p(x)=x^4+2x^3+7x^2-6x+44$ is a polynomial with real coefficients, so each complex root exists in a pair of conjugates.

Hence, $-2-i\sqrt{7}$ and $1+i\sqrt{3}$ are also roots of the given polynomial.

Thus, all the four roots of the polynomial $p(x)=x^4+2x^3+7x^2-6x+44$, are: $r_1=-2+i\sqrt{7}, r_2=-2-i\sqrt{7}, r_3=1-i\sqrt{3}, r_4=1+i\sqrt{3}$.

So, the polynomial $p(x)=x^4+2x^3+7x^2-6x+44$ can be factorized as follows:

$\{x-(-2+i\sqrt{7})\}\{x-(-2-i\sqrt{7})\}\{x-(1-i\sqrt{3})\}\{x-(1+i\sqrt{3})\}\\=(x+2-i\sqrt{7})(x+2+i\sqrt{7})(x-1+i\sqrt{3})(x-1-i\sqrt{3})\\=\{(x+2)^2+7\}\{(x-1)^2+3\}\hspace{1cm} [\because (a+b)(a-b)=a^2-b^2]\\=(x^2+4x+4+7)(x^2-2x+1+3)\\=(x^2+4x+11)(x^2-2x+4)$

Therefore, a factorization of $x^4+2x^3+7x^2-6x+44$ is $(x^2+4x+11)(x^2-2x+4)$.

To know more about factorization, refer: brainly.com/question/25829061

#SPJ9