Math II
Connect real and complex numbers, polynomials, quadratics, proof, circles, trigonometry, probability, and modeling.
- Problem types
- 786
- Practice variants
- 3,144
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Each problem type has four distinct practice variants. Open a preview to move among all four.
Add polynomials and write the sum in standard form
Add, subtract, and multiply polynomials; understand polynomial closure under these operations.
This addition is really about matching terms that have the same power of x. We’ll keep the quadratic, linear, and constant places separate, add their signed coefficients, and then read …
Preview problemSubtract polynomials and write the difference in standard form
Add, subtract, and multiply polynomials; understand polynomial closure under these operations.
The word “from” fixes the order of this subtraction, so we need to subtract the entire second polynomial from the first. We’ll turn that subtraction into addition of the opposite, …
Preview problemSimplify a sum or difference of polynomials by combining like terms and writing standard form
Add, subtract, and multiply polynomials; understand polynomial closure under these operations.
These polynomials contain different powers, so the key is to line up every degree before adding. We’ll treat a missing power as having coefficient zero, add only coefficients attached to …
Preview problemMultiply a monomial by a polynomial using the distributive property
Add, subtract, and multiply polynomials; understand polynomial closure under these operations.
This product uses the distributive property: the monomial must multiply every term inside the polynomial. For each product, we’ll multiply the numerical coefficients and add the exponents on x, then …
Preview problemMultiply two binomials and simplify the polynomial product
Add, subtract, and multiply polynomials; understand polynomial closure under these operations.
Multiplying two binomials means accounting for all four pairwise products, including the two cross-products in the middle. We’ll distribute each term of the first factor across the second, combine the …
Preview problemMultiply a binomial by a trinomial and simplify by combining like terms
Add, subtract, and multiply polynomials; understand polynomial closure under these operations.
The safest way to multiply this binomial and trinomial is to build one complete partial product from each binomial term. We’ll distribute x across all three terms, do the same …
Preview problemUse an area model to multiply polynomials and combine like terms
Add, subtract, and multiply polynomials; understand polynomial closure under these operations.
The area model organizes the same pairwise multiplication as distribution. Each cell comes from multiplying its row label by its column label, so we’ll fill the four cells in the …
Preview problemMultiply polynomials using organized partial products
Add, subtract, and multiply polynomials; understand polynomial closure under these operations.
Organized partial products let us handle the two terms in x minus three one at a time. Multiplying by x shifts every term up one degree, while multiplying by negative …
Preview problemExpand a binomial product using a special-product pattern
Add, subtract, and multiply polynomials; understand polynomial closure under these operations.
Because the same sum is multiplied by itself, this is a square-of-a-sum pattern. We’ll use a squared plus two a b plus b squared, match a with x and b …
Preview problemFind the degree and leading term of a polynomial expression
Add, subtract, and multiply polynomials; understand polynomial closure under these operations.
This problem asks us to simplify first and then interpret standard form. We’ll combine or reorder terms from greatest exponent to least, find the first nonzero term in that order, …
Preview problemDecide whether an expression is a polynomial in x
Add, subtract, and multiply polynomials; understand polynomial closure under these operations.
Classifying this expression means checking two separate polynomial requirements. We’ll expose the exponent on every x, including the zero exponent hidden in the constant term, and check that those exponents …
Preview problemWrite and simplify a polynomial expression for geometric area or volume
Add, subtract, and multiply polynomials; understand polynomial closure under these operations.
The geometry tells us which operation to use: a rectangle’s area is its length multiplied by its width. We’ll multiply the two displayed polynomial dimensions, combine the linear terms, and …
Preview problemRewrite a polynomial expression in simplified standard form by carrying out operations and combining like terms by degree
Add, subtract, and multiply polynomials; understand polynomial closure under these operations.
To put this product in standard form, every term in one binomial has to multiply every term in the other. We’ll form all four pairwise products, keep careful track of …
Preview problemWrite and solve a linear equation for a fixed fee plus a rate
Create and solve one-variable equations and inequalities, including absolute-value, linear, quadratic, simple rational, and exponential cases.
This situation has one charge paid once and another charge repeated for every visit. We’ll model the total as fixed fee plus rate times number of visits, set that expression …
Preview problemWrite and solve a linear inequality from a fixed amount and a constant rate
Create and solve one-variable equations and inequalities, including absolute-value, linear, quadratic, simple rational, and exponential cases.
The spending cap turns the fixed-fee model into an inequality instead of an equation. We’ll translate “at most” with an inclusive comparison, subtract the entry fee, divide by the positive …
Preview problemWrite and solve an absolute-value equation from a distance-from-a-target statement
Create and solve one-variable equations and inequalities, including absolute-value, linear, quadratic, simple rational, and exponential cases.
This is a distance-from-a-target situation, so absolute value captures how far the measurement is from the target without choosing a direction. We’ll set that distance equal to the given amount, …
Preview problemCreate and solve an absolute-value inequality from a distance-from-target statement
Create and solve one-variable equations and inequalities, including absolute-value, linear, quadratic, simple rational, and exponential cases.
The tolerance describes all lengths whose distance from the target is no greater than the allowed amount. We’ll express that distance with absolute value, convert the inclusive tolerance into a …
Preview problemCreate and solve a quadratic equation from an area context
Create and solve one-variable equations and inequalities, including absolute-value, linear, quadratic, simple rational, and exponential cases.
The rectangle turns the two related dimensions into a quadratic because width times length must equal the given area. We’ll write that area equation, move everything to one side and …
Preview problemCreate and solve a quadratic equation from motion context
Create and solve one-variable equations and inequalities, including absolute-value, linear, quadratic, simple rational, and exponential cases.
Reaching the ground means the height model has output zero, so the needed times are the roots of a quadratic equation. We’ll factor the model, solve for both graph intersections …
Preview problemCreate and solve a quadratic inequality from context
Create and solve one-variable equations and inequalities, including absolute-value, linear, quadratic, simple rational, and exponential cases.
Positive profit asks where the quadratic lies strictly above the zero target, not merely where it touches that line. We’ll factor to find the boundary values, determine the sign on …
Preview problemCreate and solve a rational equation where a total is divided by the unknown
Create and solve one-variable equations and inequalities, including absolute-value, linear, quadratic, simple rational, and exponential cases.
Average speed is distance divided by time, so the unknown travel time belongs in the denominator of the rate model. We’ll state that time must be positive, set the distance-over-time …
Preview problemCreate and solve a rational inequality from a constraint
Create and solve one-variable equations and inequalities, including absolute-value, linear, quadratic, simple rational, and exponential cases.
Spreading a fixed cost over more kilograms puts the positive quantity in the denominator, and “at most” creates an inclusive inequality. We’ll use the stated positive domain before clearing that …
Preview problemWrite and solve an exponential equation from a repeated-factor context
Create and solve one-variable equations and inequalities, including absolute-value, linear, quadratic, simple rational, and exponential cases.
Doubling each hour is repeated multiplication, so the elapsed time belongs in the exponent of an exponential model. We’ll set initial amount times the doubling factor to the time power …
Preview problemChoose the equation type that matches a context
Create and solve one-variable equations and inequalities, including absolute-value, linear, quadratic, simple rational, and exponential cases.
The family comes from how the balance changes, while the form comes from how that balance is compared with the target. We’ll distinguish repeated addition from repeated multiplication, then decide …
Preview problemInterpret solution candidates in context
Create and solve one-variable equations and inequalities, including absolute-value, linear, quadratic, simple rational, and exponential cases.
Algebraic candidates still have to make sense for the quantity x represents. We’ll use the contextual rule that a rectangle side length must be positive, test each listed value separately …
Preview problemWrite a one-variable equation from a geometry diagram or description
Create and solve one-variable equations and inequalities, including absolute-value, linear, quadratic, simple rational, and exponential cases.
The diagram gives two side lengths and a total area, so the governing relationship is rectangle area equals width times length. We’ll substitute the two displayed expressions into that product …
Preview problemDecide which proposed equation, inequality, or expression correctly models a context
Create and solve one-variable equations and inequalities, including absolute-value, linear, quadratic, simple rational, and exponential cases.
Before comparing the proposed models, we need the governing relationship from the context. We’ll use the fact that rectangle area multiplies two side lengths, then check each model’s operation and …
Preview problemWrite a two-variable linear equation from a fixed amount and a rate
Create equations in two or more variables and graph them with appropriate labels and scales.
This cost has a fixed part paid once and a variable part that grows by the same amount for each item. We’ll define the input as the item count and …
Preview problemWrite a quadratic equation in two variables from a verbal situation
Create equations in two or more variables and graph them with appropriate labels and scales.
The two variables play different roles: x controls the rectangle’s dimensions, while A records the resulting area. We’ll multiply length by width to connect them, and that product creates an …
Preview problemWrite a two-variable exponential equation from an initial value and a percent change
Create equations in two or more variables and graph them with appropriate labels and scales.
A fixed percent increase each year means the population is multiplied by the same growth factor repeatedly. We’ll convert the percent to a decimal, add it to one to keep …
Preview problemWrite a multi-variable formula from a measurement context
Create equations in two or more variables and graph them with appropriate labels and scales.
A rectangular prism’s volume comes from extending its rectangular base through a height. We’ll multiply length and width for the base area, multiply that result by height, and check that …
Preview problemGraph a created equation in context
Create equations in two or more variables and graph them with appropriate labels and scales.
The graph has to communicate both the linear rule and the item-count context. We’ll place items on the horizontal axis and cost on the vertical axis, use the fixed fee …
Preview problemChoose an appropriate graph scale
Create equations in two or more variables and graph them with appropriate labels and scales.
A useful scale must include every required endpoint without wasting so much space that the data are compressed. We’ll compare each frame’s minimum and maximum with the stated time and …
Preview problemIdentify domain and range restrictions from context
Create equations in two or more variables and graph them with appropriate labels and scales.
The equation alone allows real inputs, but the context restricts x because purchased items are counted. We’ll build the meaningful domain from nonnegative whole-number counts, substitute those inputs into the …
Preview problemInterpret a point on the graph of an equation in context
Create equations in two or more variables and graph them with appropriate labels and scales.
A point on this graph is an input-output pair, so the first coordinate uses item units and the second uses dollar units. We’ll substitute the input into the full cost …
Preview problemDecide whether two equations represent the same relationship
Create equations in two or more variables and graph them with appropriate labels and scales.
Different-looking equations can still describe the same relationship if one can be rewritten exactly as the other. We’ll distribute x across the full binomial in the factored area form and …
Preview problem