4
1xy
Usually simple substitutions enable such expressions to be seen as quadratic expressions. The substitutions x = a2 and y = b2 give a4 + b4 - 7a2b2 = x2 - 7xy + y2 which does not have any rational factors. Consequently, the quartic in a and b does not have rational factors.
All of these are monomials. The binomials are algebraic expressions consisting of two terms. A term is a constant, a variable, or a combination of constants and variables. 2y, 2x, y, and 2xy consisting of only one term, so they are monomials. Examples of binomials: 2 + 5, 2x + 3, 3x + 7y, 2xyz + 7xy,...
The factors of 7xyz would include 1, 7, x, y, z, and any combination of these factors. Factors are numbers or variables that can be multiplied together to give the original number or expression. In this case, the factors of 7xyz would be all the numbers and variables that divide evenly into 7xyz without leaving a remainder.
4
5
(xy - 7)(x^2y^2 + 7xy + 49)
The GCF is 7xy^3
-9x^2 - 15x
5x(7xy + 6x + 8)
A quadratic. For example, x^2 + 3x - 7xy - 12y - 25 = 0
-7xy(x4y2 + 6x2y - 5)
gcf(35x², 7xy², 21x²y²) = 7x 35x² = 5 times 7 times x² 7xy² = 7 times x times y² 21x²y² = 3 times 7 times x² times y² hcf = 7 times x = 7x (I've written the multiplication as times instead of using the × symbol to avoid confusion with the unknown x.)
It means "1" is the magic number, like if you were to have a math problem that looked like this:7xy-xy+3xIt means use the "magic one" to combine 7xy and 1xyIf that doesn't make sense you should probably state the question like this:What does the "Magic 1" mean in math?
(xy + 7)(x^2y^2 - 7xy + 49)
1xy