Factor out completely: 4a^2c^2-(a^2-b^2 c^2)^2

The expression 4a^2c^2 - (a^2-b^2+c^2)^2 has to be factored.

4a^2c^2 - (a^2 - b^2 + c^2)^2

=> (2ac)^2 - (a^2 - b^2 + c^2)^2

=> (2ac - a^2 + b^2 - c^2)(2ac + a^2 - b^2 + c^2)

=> (b^2 - (a^2 - 2ac + c^2))((a^2 + 2ac + c^2) - b^2)

=> (b^2 - (a - c)^2)((a + c)^2 - b^2)

=> (b - a + c)(b + a - c)(a + b + c)(a - b + c)

The factorized form of 4a^2c^2 - (a^2-b^2+c^2)^2 is (b - a + c)(b + a - c)(a + b + c)(a - b + c)

yes

yes yes yes

the correct option is:

b. a value of the variable that makes the polynomial equal to zero.

the root or roots of a polynomial are the values of the variable that makes the polynomial equal to 0. we can find the roots of a polynomial making the expression equal to 0 and then, isolating the variable (it it has just one variable).

for example, given the following polynomial:

the root or zero of the polynomial, can be found by making the expression equal to 0, and then, isolating the variable, so, finding the root we have:

therefore, we have that the root of the polynomial is equal to 2.

we must remember that the number of roots that a polynomial can have, will depend on the degree of the polynomial, a first degree polynomial have will have one single root, a second degree polynomial (quadratic expression) may have two roots and so.

hence the correct option is:

b. a value of the variable that makes the polynomial equal to zero.

have a nice day!