You can find any term in an arithmetic or geometric list and add a whole run of terms quickly.
Weeks 13-14
Algebra II course units
Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.
01
Polynomial functions
Weeks 1-2
You can factor higher-degree polynomials, find all their zeros, and predict how the ends of the graph behave. You can also divide one polynomial by another.
Use synthetic division to test whether x = 2 is a root.
Factor x cubed - 8 as a difference of two cubes.
Predict that an odd-degree polynomial with a positive lead falls on the left and rises on the right.
Worked example
Is x = 2 a root of x cubed - 3x squared + 4?
Substitute: 8 - 3(4) + 4
8 - 12 + 4
Answer0, so yes, x = 2 is a root
Most common mistakeMultiplying before squaring: reading 3x squared at x = 2 as (3 x 2) squared = 36, so the total looks like -24 and the root is missed.
You can simplify fractions that contain variables, solve equations built from them, and find where the graph shoots up or flattens out. You can also spot answers that have to be thrown away.
Find the vertical asymptote of f(x) = (x + 1)/(x - 4).
Add 1/x + 1/(x + 2) over a common denominator.
Check each answer against the denominator and reject any that make it zero.
Worked example
Solve 3/(x - 2) = 1
Multiply both sides by (x - 2): 3 = x - 2
x = 5
Answerx = 5 is allowed, since 5 - 2 = 3 is not zero
Most common mistakeKeeping an answer that breaks the denominator: solving x/(x - 3) = 3/(x - 3) and reporting x = 3, even though x = 3 makes both fractions undefined and the equation has no solution.
You can simplify roots, combine them, and solve equations that hide the variable under a root sign. You can also rewrite any root as a fractional exponent.
Simplify the square root of 50 into 5 times the square root of 2.
Rationalize the denominator of 6 divided by the square root of 3.
Solve a root equation and then check for answers that do not really work.
Worked example
Solve the square root of (2x + 1) = 5
Square both sides: 2x + 1 = 25
2x = 24
Answerx = 12 (check: the square root of 25 is 5)
Most common mistakeSquaring term by term: turning the equation into 2x + 1 = 5 and answering x = 2, which fails the check because the square root of 5 is only about 2.24.
You can add, multiply and divide numbers built from i, the square root of -1. You can also finish quadratics that have no real answers.
Simplify the square root of -49 into 7i.
Multiply (3 + 2i)(1 - 4i) and replace i squared with -1.
Divide by a complex number using its conjugate.
Worked example
Multiply (3 + 2i)(1 - 4i)
3 - 12i + 2i - 8i squared
i squared = -1, so -8i squared = +8
Answer11 - 10i
Most common mistakeTreating i squared as 1: getting 3 - 10i - 8 = -5 - 10i instead of 11 - 10i, a sign slip that moves the answer to the wrong half of the plane.
You can write and use a rule where an amount grows or shrinks by a percent every period. You can also read the starting value and the rate straight off the formula.
Write 6% yearly growth on a starting amount of 500 as P = 500(1.06) to the power t.
Tell growth from decay by whether the base sits above or below 1.
Find the value of a 20,000 car after four years of 15% yearly decline.
Worked example
A population of 500 grows 6% each year. What is it after 10 years?
P = 500 x 1.06 to the power 10
1.06 to the power 10 = 1.7908
AnswerAbout 895
Most common mistakeUsing simple interest instead of a power: computing 500 + 500(0.06)(10) = 800, which misses the growth that compounds on earlier growth.
You can rewrite an exponent question as a logarithm, break logs apart, and solve for a variable stuck up in the exponent. You can also use logs on real growth problems.
Rewrite 2 to the fifth = 32 as log base 2 of 32 = 5.
Expand log(xy squared) into log x + 2 log y.
Solve 3 to the power x = 40 with logarithms.
Worked example
Solve 3 to the power x = 40
Take logs of both sides: x log 3 = log 40
x = 1.6021 / 0.4771
Answerx is about 3.358
Most common mistakeDividing the numbers rather than their logs: computing 40 / 3 = 13.3 and calling that x, when 3 to the power 13.3 is over a million.
You can find any term in an arithmetic or geometric list and add a whole run of terms quickly. You can also tell which of the two types you are looking at.
Find the 20th term of 3, 7, 11, 15, and so on.
Spot a geometric sequence by dividing each term by the one before it.
Add the first 10 terms of an arithmetic sequence using average times count.
Worked example
Find the 20th term of 3, 7, 11, 15, ...
First term a = 3, common difference d = 4
a20 = 3 + (20 - 1) x 4 = 3 + 76
Answer79
Most common mistakeMultiplying by n instead of n - 1: computing 3 + 20 x 4 = 83, one full step past the term you wanted.
Before timing yourself, check whether you can explain Polynomial functions from a blank page. Then connect it to Rational functions. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.
Use tools to verify, not to choose the method for you
The Algebra calculator can test calculations and representations used in Algebra II. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.
Know when to move to the full test
Move from Algebra II practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.
Before and after the syllabus
Learn the ideas, then practise them
The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.
Start with Polynomial functions if you are following the full sequence. If that unit feels automatic, open the Algebra II practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.
How do I know I am ready for the course test?
You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Algebra II test only after you can correct practice errors from a blank page.
Which calculator supports this course?
The Algebra calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.
