I would like to know how to combine like terms with different coefficients and exponents. On my homework, the problems have the variable (x) but there are no other variables in the problems.

Like terms are where the exponents and the variable are the same but the coefficients may be different. Constants are all considered to be like terms.

The same variable with different exponents can only be combined by factorisation, the variable with the lowest exponent usually being a common factor. Factorisation can also be used when two coefficients have a common factor. You can't combine different variables, even if they have the same exponent.

Factorisation can be used when you have products of different variables.

EXAMPLE

16+3x^3+5x^2+2x^3-y^2-x+4y+8x+y+6y^2-2x+9-25 can be simplified:

x^3 term: 3x^3+2x^3 is 5x^3

x^2 term: 5x^2 only

y^2 term: -y^2+6y^2 is 5y^2

x term: -x+8x-2x is 5x

y term: 4y+y is 5y

constant: 16+9-25 is 0

So we have: 5x^3+5x^2+5y^2+5x+5y. But 5 is a common factor: 5(x^3+x^2+y^2+x+y).

Further factorisation is possible but looks clumsy: 5(x(x^2+x+1)+y(y+1))

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