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.
Add higher-degree polynomials by combining like terms
Add, subtract, and multiply polynomials beyond quadratic cases.
This is a polynomial addition problem, so the structure comes from matching terms with exactly the same power of x. We’ll group the fourth-power terms, squared terms, and constants, then …
Preview problemSubtract higher-degree polynomials by combining like terms
Add, subtract, and multiply polynomials beyond quadratic cases.
The main challenge is the subtraction sign in front of the second polynomial, because it applies to every term inside those parentheses. We’ll rewrite the subtraction as adding the opposite, …
Preview problemAdd or subtract multivariable polynomials by combining like terms
Add, subtract, and multiply polynomials beyond quadratic cases.
This problem combines two ideas: distributing the subtraction across the second polynomial and deciding which multivariable terms are truly alike. We’ll reverse both signs, then combine only terms whose x …
Preview problemMultiply a monomial by a polynomial using the distributive property
Add, subtract, and multiply polynomials beyond quadratic cases.
This product uses distribution: the monomial outside the parentheses must multiply every term inside. For each product, we’ll multiply the numerical coefficients and add exponents on matching x factors, while …
Preview problemMultiply a binomial by a polynomial by distributing both terms
Add, subtract, and multiply polynomials beyond quadratic cases.
The binomial creates two complete partial products, one from x and one from 2. We’ll distribute each of those terms across the entire cubic polynomial, line up results by degree, …
Preview problemMultiply higher-degree polynomials and combine like terms
Add, subtract, and multiply polynomials beyond quadratic cases.
This multiplication produces three partial products because every term in the first factor must multiply both terms in the second. We’ll write those products separately, use exponent addition for each …
Preview problemMultiply a binomial and a trinomial with a box or area model
Add, subtract, and multiply polynomials beyond quadratic cases.
The box model organizes the multiplication so every row label meets every column label exactly once. We’ll fill all six cells with those term products, then collect cells that have …
Preview problemExpand a polynomial using a special-product identity
Add, subtract, and multiply polynomials beyond quadratic cases.
This expression is a binomial square, so a special-product identity gives a shorter route than distributing from scratch. We’ll treat x squared as the first part and 3 as the …
Preview problemSimplify a polynomial and identify its degree and leading coefficient
Add, subtract, and multiply polynomials beyond quadratic cases.
The requested features all depend on the polynomial after like terms have been combined. We’ll simplify first, order the surviving terms by descending power, and then read three pieces from …
Preview problemDetermine polynomial end behavior from the leading term
Add, subtract, and multiply polynomials beyond quadratic cases.
End behavior is controlled by the leading term because its highest power dominates for large inputs. We’ll use the odd exponent to determine that the two ends have opposite signs, …
Preview problemModel a context with polynomial multiplication and simplify the expression
Add, subtract, and multiply polynomials beyond quadratic cases.
This is a volume problem before it is a polynomial problem: the three dimension expressions must be multiplied, not added. We’ll form the product of all three dimensions, multiply two …
Preview problemDecide whether two polynomial expressions are equivalent by rewriting them
Add, subtract, and multiply polynomials beyond quadratic cases.
Equivalent polynomial expressions must produce the same simplified form for every x. We’ll expand the squared binomial with the middle cross term included, distribute the outside x squared, and compare …
Preview problemSimplify a nested polynomial expression step by step
Add, subtract, and multiply polynomials beyond quadratic cases.
The nesting determines the order: simplify each product inside its own grouping before dealing with the outside subtraction. We’ll expand both products, treat the bracket as one whole polynomial, reverse …
Preview problemEvaluate a polynomial at a given value to find the remainder
Apply the Remainder Theorem to connect p(a), remainders, and factors x-a.
Finding the remainder here is the same as evaluating the polynomial at the given input. We’ll replace every x by 2, evaluate each term with its original sign, and add …
Preview problemUse the Remainder Theorem to find a remainder from a divisor x-a
Apply the Remainder Theorem to connect p(a), remainders, and factors x-a.
The divisor already has the form x minus a, so the Remainder Theorem turns division into one polynomial evaluation. We’ll identify a as positive 2, substitute that value into every …
Preview problemUse synthetic division to find the remainder from a divisor x-a
Apply the Remainder Theorem to connect p(a), remainders, and factors x-a.
Synthetic division compresses long division into a repeated multiply-and-add process. We’ll use positive 2 from the divisor, keep a zero placeholder for the missing squared term, bring down the first …
Preview problemUse polynomial long division to find the remainder from a linear divisor
Apply the Remainder Theorem to connect p(a), remainders, and factors x-a.
Polynomial long division works by canceling the current leading term over and over. We’ll first insert a zero x term to keep the powers aligned, then repeat divide, multiply back, …
Preview problemUse the Factor Theorem to decide whether x-a is a factor
Apply the Remainder Theorem to connect p(a), remainders, and factors x-a.
This is a factor test, and the Factor Theorem reduces it to one evaluation. The factor x minus 2 tells us to evaluate p at positive 2; if that value …
Preview problemFind a missing coefficient that makes x-a a factor
Apply the Remainder Theorem to connect p(a), remainders, and factors x-a.
The unknown coefficient is constrained by the requirement that x minus 2 divide the polynomial evenly. We’ll translate that factor statement into p of 2 equals zero, substitute 2 everywhere …
Preview problemFind a missing coefficient from remainder information using the Remainder Theorem
Apply the Remainder Theorem to connect p(a), remainders, and factors x-a.
The stated remainder gives an equation for the unknown coefficient through the Remainder Theorem. Since the divisor is x minus 2, we’ll set p of 2 equal to 7, substitute …
Preview problemConnect polynomial evaluations, zeros, factors, and x-intercepts
Apply the Remainder Theorem to connect p(a), remainders, and factors x-a.
The statement p of 3 equals zero carries four equivalent mathematical meanings. We’ll keep the same input 3 while translating it into a zero, the opposite-sign linear factor x minus …
Preview problemUse values of p(a) to identify linear factors x-a
Apply the Remainder Theorem to connect p(a), remainders, and factors x-a.
The table matters only at rows where the output p of x is zero. We’ll collect every such input, call each one a, and convert it to the factor x …
Preview problemUse the Remainder Theorem to find a remainder or test whether x-a is a factor
Apply the Remainder Theorem to connect p(a), remainders, and factors x-a.
The seventh degree looks intimidating, but the divisor x minus 1 makes long division unnecessary. We’ll use the Remainder Theorem to identify the remainder as p of positive 1, substitute …
Preview problemInterpret a polynomial-division remainder as a contextual function value
Apply the Remainder Theorem to connect p(a), remainders, and factors x-a.
The remainder represents the revenue model’s output at the input encoded by the divisor. We’ll match x minus 5 to x minus a, connect the given remainder to R of …
Preview problemUse the Remainder Theorem to test possible rational roots
Apply the Remainder Theorem to connect p(a), remainders, and factors x-a.
Each listed number is only a possible root until evaluation confirms it. We’ll substitute every candidate separately and keep precisely those for which p of a equals zero; by the …
Preview problemFind zeros from a factored polynomial by setting each factor equal to zero
Identify zeros from factorizations and use them to sketch polynomial graphs.
Factored form turns zero-finding into three small linear equations. Since a product equals zero when any factor is zero, we’ll set each factor separately equal to zero and solve, including …
Preview problemEnter each polynomial zero with its multiplicity
Identify zeros from factorizations and use them to sketch polynomial graphs.
Each repeated factor carries two pieces of information: the value that makes its base zero and the exponent that counts how many times it repeats. We’ll solve the base of …
Preview problemDetermine how a polynomial graph behaves at a zero from its multiplicity
Identify zeros from factorizations and use them to sketch polynomial graphs.
Graph behavior at a zero comes from the parity of its multiplicity. We’ll locate the zero on the x-axis, classify multiplicity 1 as odd, and use the polynomial’s sign on …
Preview problemDetermine the end behavior of a polynomial written in factored form
Identify zeros from factorizations and use them to sketch polynomial graphs.
We do not need to expand the whole polynomial to find its end behavior. We’ll multiply only the leading parts of the factors: their degrees add to give the total …
Preview problemExtract zero behavior and end behavior from factored form
Identify zeros from factorizations and use them to sketch polynomial graphs.
Factored form is a blueprint for both the local and global shape of the graph. We’ll read each zero and exponent, use multiplicity parity to decide what happens at that …
Preview problemMatch a factored polynomial to its graph using zeros, multiplicities, and end behavior
Identify zeros from factorizations and use them to sketch polynomial graphs.
The factorization gives three independent filters for the graph. First locate each zero; then use the exponent on its factor to predict a crossing or a bounce. Finally, combine the …
Preview problemWrite the unique least-degree factored polynomial with a specified leading coefficient
Identify zeros from factorizations and use them to sketch polynomial graphs.
Build the polynomial from the graph one intercept at a time. A crossing needs an odd multiplicity and a touch needs an even multiplicity; because the problem asks for least …
Preview problemFind the y-intercept of a polynomial written in factored form
Identify zeros from factorizations and use them to sketch polynomial graphs.
An intercept on the y-axis occurs where the input is zero, so this is an evaluation problem rather than a zero-finding problem. Substitute zero into the factored expression without expanding …
Preview problemUse a known zero to factor a numeric polynomial and identify graph features
Identify zeros from factorizations and use them to sketch polynomial graphs.
Use the known zero to unlock the rest of the polynomial. The Factor Theorem turns that zero into a linear divisor; synthetic division lowers the degree, making the remaining quotient …
Preview problemUse the degree of a polynomial to find the greatest possible number of turning points
Identify zeros from factorizations and use them to sketch polynomial graphs.
The degree controls how much a polynomial can turn, but the bound is a maximum rather than a guarantee. A degree-n polynomial can have at most n minus one turning …
Preview problemDistinguish real and complex zeros in a factorization for graph sketching
Identify zeros from factorizations and use them to sketch polynomial graphs.
Separate algebraic zeros from visible x-intercepts. Solve each factor over the complex numbers, classify the resulting zeros as real or nonreal, and remember that only real zeros can appear as …
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