Working with radical and rational equations
A radical equation is one that starts out with a square root, cube root, or some other root and gets changed into another form to make the solving process easier. A rational equation is one that involves a fractional expression — usually with a polynomial in the numerator and denominator. Avoid these mistakes when working with radical or rational equations:
-
Forgetting to check for extraneous solutions
-
Squaring a binomial incorrectly when squaring both sides to get rid of the radical
-
Distributing correctly when writing equivalent fractions using a common denominator
-
Eliminating solutions that create a 0 in the denominator
Polynomial functions and equations
In Algebra II, a polynomial function is one in which the coefficients are all real numbers, and the exponents on the variables are all whole numbers. A polynomial whose greatest power is 2 is called a quadratic polynomial; if the highest power is 3, then it’s called a cubic polynomial. A highest power of 4 earns the name quartic (not to be confused with quadratic), and a highest power of 5 is called quintic.
When solving polynomial functions and equations, don’t let these common mistakes trip you up:
-
Forgetting to change the signs in the factored form when identifying x-intercepts
-
Making errors when simplifying the terms in f(–x) applying Descartes’ rule of sign
-
Not changing the sign of the divisor when using synthetic division
-
Not distinguishing between curves that cross from those that just touch the x-axis at an intercept
-
Graphing the incorrect end-behavior on the right and left of the graphs
Systems of linear equations
In Algebra II, a linear equation consists of variable terms whose exponents are always the number 1. When you have two variables, the equation can be represented by a line. With three terms, you can draw a plane to describe the equation. More than three variables is indescribable, because there are only three dimensions. When you have a system of linear equations, you can find the values of the variables that work for all the equations in the system — the common solutions. Sometimes there’s just one solution, sometimes many, and sometimes there’s no solution at all.
When solving systems of linear equations, watch out for these mistakes:
-
Forgetting to change the signs in the factored form when identifying x-intercepts
-
Making errors when simplifying the terms in f(–x) applying Descartes’ rule of sign
-
Not changing the sign of the divisor when using synthetic division
-
Not distinguishing between curves that cross from those that just touch the x-axis at an intercept
-
Graphing the incorrect end-behavior on the right and left of the graphs
