Math III
Go further with polynomial and rational expressions, advanced functions, trigonometry, geometric modeling, and statistical inference.
- Problem types
- 641
- Practice variants
- 2,564
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Each problem type has four distinct practice variants. Open a preview to move among all four.
Choose the best method to rewrite a rational expression as quotient plus remainder over the divisor
Rewrite rational expressions using inspection, polynomial division, or technology.
Determine the division method by matching the givens to each method’s requirements. Here the divisor is monic and linear, and the polynomial’s ordered coefficients are available; also check whether a …
Preview problemVerify a rational-expression rewrite using quotient and remainder
Rewrite rational expressions using inspection, polynomial division, or technology.
Verify a rational rewrite by reversing the division. Identify the divisor, quotient, and remainder, multiply divisor by quotient, and then add the remainder; the result must equal the original numerator …
Preview problemSimplify a factored rational expression without losing holes
Rewrite rational expressions using inspection, polynomial division, or technology.
A canceled factor simplifies values only where the original expression was already defined. Record the denominator zero first, factor the numerator completely, and cancel the common factor on the allowed …
Preview problemInterpret quotient trend and vanishing remainder correction
Rewrite rational expressions using inspection, polynomial division, or technology.
Separate the model into a quotient trend and a remainder correction. The correction’s magnitude tells how far the function is from the trend, its sign tells which side of the …
Preview problemFind a missing coefficient using the remainder when dividing by a linear binomial
Rewrite rational expressions using inspection, polynomial division, or technology.
The Remainder Theorem turns a division condition into one evaluation equation. Rewrite the divisor as x minus a to identify the correct signed input, set the polynomial’s value at that …
Preview problemDecide rational-rewrite equivalence using algebra and original domains
Rewrite rational expressions using inspection, polynomial division, or technology.
Equivalence of rational expressions requires two matches: the same value formula and the same domain. Recombine a quotient-plus-remainder form over its shared denominator, simplify the numerator, and compare it with …
Preview problemMultiply rational expressions and preserve original-domain restrictions
Add, subtract, multiply, and divide rational expressions as a closed system like rational numbers.
For a product of rational expressions, preserve the domain before simplifying. List zeros of every original denominator, factor all polynomials, and view the entire multiplication as one numerator over one …
Preview problemDivide rational expressions with denominator and zero-divisor restrictions
Add, subtract, multiply, and divide rational expressions as a closed system like rational numbers.
Dividing rational expressions creates two kinds of restrictions before any reciprocal is taken. Exclude zeros of all original denominators, and also exclude inputs that make the entire divisor rational expression …
Preview problemAdd rational expressions that already have a common denominator, then simplify and state restrictions
Add, subtract, multiply, and divide rational expressions as a closed system like rational numbers.
When rational expressions already share a denominator, treat them like ordinary fractions with that same denominator. Record its zeros, keep the denominator unchanged, and combine only the numerators according to …
Preview problemSubtract rational expressions over a common denominator with restrictions
Add, subtract, multiply, and divide rational expressions as a closed system like rational numbers.
With a shared denominator, subtraction acts on the entire second numerator. Keep the denominator and its original restriction, place both numerators in parentheses, and distribute the negative sign through every …
Preview problemAdd unlike rational expressions with an LCD and domain restrictions
Add, subtract, multiply, and divide rational expressions as a closed system like rational numbers.
For unlike denominators, factor first and build the least common denominator from every required factor at its highest power. Record all original denominator zeros, then multiply each fraction by only …
Preview problemSubtract unlike rational expressions with an LCD and restrictions
Add, subtract, multiply, and divide rational expressions as a closed system like rational numbers.
Build equivalent fractions over the least common denominator before subtracting. Each numerator must receive the factor missing from its own denominator, and the adjusted second numerator stays grouped so the …
Preview problemSimplify a complex rational expression with all restriction sources
Add, subtract, multiply, and divide rational expressions as a closed system like rational numbers.
A complex fraction has restrictions hidden at two levels. First require every inner denominator to be nonzero, then check whether the entire expression in the main denominator can equal zero. …
Preview problemState restrictions before simplifying a rational expression
Add, subtract, multiply, and divide rational expressions as a closed system like rational numbers.
Restrictions belong to the original expression, so determine them before factoring or cancellation. Set each original denominator equal to zero and exclude those solutions; zeros that occur only in numerators …
Preview problemModel rate and work contexts with algebraic and physical domains
Add, subtract, multiply, and divide rational expressions as a closed system like rational numbers.
Convert each completion time to a rate before combining anything: one job completed in t hours corresponds to one over t jobs per hour. Rates add when workers operate together, …
Preview problemDecide whether two rational expressions are equivalent
Add, subtract, multiply, and divide rational expressions as a closed system like rational numbers.
To compare rational expressions, separate agreement of values from agreement of domains. Simplify each formula on its allowed inputs, then list every original excluded value and test any input that …
Preview problemAdd rational expressions by factoring, simplifying, and stating denominator restrictions
Add, subtract, multiply, and divide rational expressions as a closed system like rational numbers.
Factor the denominators before constructing the least common denominator, and record every original denominator zero. Rewrite each fraction by multiplying its numerator and denominator by exactly the missing factors, then …
Preview problemSimplify a rational operation and state all original restrictions
Add, subtract, multiply, and divide rational expressions as a closed system like rational numbers.
Use a stable order for any rational operation: record original restrictions, perform the operation, factor completely, cancel whole common factors, and then restate the restrictions. Terms joined by addition or …
Preview problemWrite and solve a polynomial equation from a context
Create and solve one-variable equations and inequalities from contexts using all studied expression types, including simple root functions.
Treat the three related dimensions as factors in one product before doing any algebra. Zero form then turns the context into a polynomial root problem, but the roots are only …
Preview problemWrite and solve a one-variable rational equation from a context
Create and solve one-variable equations and inequalities from contexts using all studied expression types, including simple root functions.
In a work problem, completion times do not add; their reciprocals are the rates that combine. Write the rate equation while recording that time must be positive, then clear denominators …
Preview problemWrite and solve a radical equation from a context
Create and solve one-variable equations and inequalities from contexts using all studied expression types, including simple root functions.
Because the radical is isolated and both sides are nonnegative, squaring is a valid way to expose the equation underneath. When that equation determines a square, remember that opposite inputs …
Preview problemWrite and solve an exponential equation from a context
Create and solve one-variable equations and inequalities from contexts using all studied expression types, including simple root functions.
A repeated percent increase becomes a multiplier greater than one, not the percent written as a decimal by itself. After setting the growth expression equal to the target, divide out …
Preview problemWrite and solve a one-variable equation from a logarithmic or exponential formula
Create and solve one-variable equations and inequalities from contexts using all studied expression types, including simple root functions.
First isolate the logarithm carefully, since the negative sign belongs outside it and changes the logarithm's output. Then use the inverse relationship between a base-ten logarithm and a power of …
Preview problemWrite and solve an absolute-value equation or inequality from a context
Create and solve one-variable equations and inequalities from contexts using all studied expression types, including simple root functions.
Read “within” as a distance condition: the measurement may lie on either side of the target, but no farther than the stated tolerance. An absolute-value bound captures both sides at …
Preview problemWrite and solve a one-variable root-function inequality from a context
Create and solve one-variable equations and inequalities from contexts using all studied expression types, including simple root functions.
Start with the square root's domain, since any value making the radicand negative is excluded before solving begins. On that domain, both sides are nonnegative, so squaring preserves the inequality …
Preview problemChoose the one-variable model type that best fits a context
Create and solve one-variable equations and inequalities from contexts using all studied expression types, including simple root functions.
Classify a model by the operation that repeats over equal intervals. A fixed amount added each year points to constant additive change, while a fixed percent makes each new value …
Preview problemDetermine which solutions to a modeled equation are valid in context
Create and solve one-variable equations and inequalities from contexts using all studied expression types, including simple root functions.
An equation can produce algebraic candidates that the story itself rules out. Translate “elapsed time after the timer starts” into a nonnegative domain, then test each candidate against that condition. …
Preview problemWrite and solve a contextual constraint in one variable, including the feasible domain
Create and solve one-variable equations and inequalities from contexts using all studied expression types, including simple root functions.
Use the net to name all three folded dimensions before writing the volume equation. The same dimensions also impose the feasible cut-size domain, so record that domain before solving the …
Preview problemCreate an equation from a model and solve for the input value
Create and solve one-variable equations and inequalities from contexts using all studied expression types, including simple root functions.
Read the table in two layers: the entry at input zero gives the starting value, while successive output ratios reveal what repeats. Those features determine the model before the target …
Preview problemSelect a one-variable model family from a defining structural feature
Create and solve one-variable equations and inequalities from contexts using all studied expression types, including simple root functions.
Focus on how the population changes, not merely on the fact that it increases. A fixed percent of the current amount creates the same multiplier each year, whereas a fixed …
Preview problemUse technology to solve a one-variable model and report the viable solution
Create and solve one-variable equations and inequalities from contexts using all studied expression types, including simple root functions.
Turn the equation into a zero-finding problem and search only in the positive region required by the context. A sign change brackets a crossing, and technology can refine that crossing …
Preview problemDetermine domain restrictions before solving a modeled equation
Create and solve one-variable equations and inequalities from contexts using all studied expression types, including simple root functions.
Domain restrictions come from the original expression, before any algebra changes its appearance. For a rational equation, locate every denominator and find the inputs that make one equal zero. Record …
Preview problemWrite an equation in two variables from a context
Create equations in two or more variables, graph them, and interpret relationships with labels and scales.
Begin with the geometric quantity that is held fixed, then connect it to the two named dimensions. Rectangle area comes from multiplying perpendicular side lengths, so the relationship must preserve …
Preview problemWrite a rational equation in two variables from a context
Create equations in two or more variables, graph them, and interpret relationships with labels and scales.
Anchor the model in distance equals speed times time, with the trip distance fixed. Solving that relationship for time puts speed in the denominator, which captures the inverse tradeoff: higher …
Preview problemWrite a radical equation in two variables from a context
Create equations in two or more variables, graph them, and interpret relationships with labels and scales.
Read the coordinate differences as perpendicular leg lengths, not quantities to add directly. The Pythagorean relationship combines their squares to produce the square of the distance. Taking the principal square …
Preview problemWrite an exponential or logarithmic equation in two variables from a context
Create equations in two or more variables, graph them, and interpret relationships with labels and scales.
Separate the two roles in the context: the starting amount sets the value at time zero, while the percent increase determines a yearly multiplier. Repeating that multiplier places elapsed years …
Preview problem