Question

factor a^3+b^2+c^2 over a^2+b+c no line between them, no signs of operation

Answer 1

Factor $a^3 + b^2 + c^2$ over $a^2 + b + c$ no line between them, no signs of operation.

Explanation

Factoring is the opposite process of multiplying polynomials, in order to return to a unique string of polynomials of lesser degree whose product is the original polynomial, is the simplest way to solve equations of higher degree. When we factor a polynomial, we are looking for simpler polynomials that can be multiplied together to give us the polynomial that we started with. If all of the terms in a polynomial contain one or more identical factors, combine those similar factors into one monomial, called the greatest common factor, and rewrite the polynomial in factored form.

In our case, $a^3 + b^2 + c^2$ we can divide on $a^2 + b + c$

Thus we can write, $(a^2 + b + c)(a + b + c) + (-ab - ac - a^2b - 2bc - a^2c) =$

$$= (a ^ {2} + b + c) (a + b + c) + (- a ^ {2} b - a ^ {2} c - 2 b c - a b - a c)$$

Answer: $(a^2 + b + c)(a + b + c) + (-a^2b - a^2c - 2bc - ab - ac)$

Check:

$$\begin{array}{l} (a ^ {2} + b + c) (a + b + c) + (- a b - a c - a ^ {2} b - b c - a ^ {2} c - b c) = (a ^ {3} + a ^ {2} b + a ^ {2} c + a b + \\ + b ^ {2} + b c + a c + b c + c ^ {2} - a b - a c - a ^ {2} b - b c - a ^ {2} c - b c) = (a ^ {3} + + b ^ {2} + c ^ {2}) \\ \end{array}$$

If we consider the second polynomial $a^2 + b + c$ , we can provide a solution in the following way - can be expressed as the sum of the squares:

$$a ^ {2} + b + c = \left(a + \left(\sqrt {b + c}\right)\right) \left(a + \left(\sqrt {b + c}\right)\right) - 2 a \sqrt {b + c}$$

Also we can check:

$$\begin{array}{l} \left(a + \left(\sqrt {b + c}\right)\right) \left(a + \left(\sqrt {b + c}\right)\right) - 2 a \sqrt {b + c} = \left(a ^ {2} + a \sqrt {b + c} + a \sqrt {b + c} + \sqrt {b + c} \times \right. \\ \sqrt {b + c} - 2 a \sqrt {b + c}) = (a ^ {2} + 2 a \sqrt {b + c} + b + c - 2 a \sqrt {b + c}) = (a ^ {2} + b + c) \\ \end{array}$$

Answer:

$$a^{2} + b + c = \left(a + (\sqrt{b + c})\right) \left(a + (\sqrt{b + c})\right) - 2a\sqrt{b + c}$$