Review polynomial vocabulary, operations, evaluation, composition, division, zeros, and graph behavior.
20 mixed polynomial questions with worked explanations and error analysis.
Polynomial questions can look unrelated because one asks for degree, another for a value, another for a remainder, and another about a graph. The unifying skill is method recognition: identify the polynomial structure first, then apply the rule that belongs to it. This guide connects vocabulary, operations, evaluation, composition, division, zeros, and graph behavior.
Write a polynomial in standard form, from highest exponent to lowest. That ordering exposes the degree and leading coefficient and makes later operations easier to organize.
The degree of a nonzero polynomial is its greatest exponent after like terms have been combined.
Addition, subtraction, and multiplication all use exponents, but not in the same way. The key distinction is simple: addition combines like terms; multiplication combines factors.
Distribute subtraction signs, align like powers, and combine coefficients only when the variable parts match.
(3x^2)(-4x^5)=-12x^7 The exponent rule belongs to multiplication, not addition.
Distribute systematically, then combine like terms. The degree of a nonzero product equals the sum of factor degrees.
A cubic times a quartic has degree seven. This is a fast structural check on an expanded answer.
deg(PQ)=deg(P)+deg(Q)A minus sign outside parentheses changes every sign inside before like terms are combined.
A(x)-B(x)=A(x)+(-1)B(x)Evaluation replaces every occurrence of the variable with one input. Composition makes one function's output become another function's input. Parentheses matter because the entire input must replace each x-slot.
For P(a), substitute a everywhere, then use the order of operations.
P(a)If the input is multi-term or negative, substitute it as a grouped expression.
P(t-2)For f(g(x)), replace every x in f by the complete expression g(x).
f(g(x))Polynomial division splits a dividend into divisor × quotient + remainder. When the divisor is linear, the Remainder Theorem can replace a full division if the question asks only for the remainder.
If P(x) is divided by x-a, the remainder is P(a).
A zero is an input that makes the polynomial equal to zero. The same fact can be written as an evaluation statement, a root, a factor, or—when the zero is real—an x-intercept.
If P(a)=0, division by x-a has remainder zero, so x-a is a factor.
Repeated factors can make a graph cross an axis differently from a simple zero.
A polynomial may have complex zeros, so the number of real x-intercepts can be smaller than its degree.
For very large positive or negative x, the leading term dominates. That makes degree parity and leading-coefficient sign the fastest clues for end behavior. Degree also limits the number of turns.
Memorize the relationship and the signal that tells you when to use it.
Use only when variable powers match exactly.
Multiply coefficients; add exponents.
For nonzero polynomials.
Use when only the remainder is needed.
Links evaluation, roots, and division.
The inner expression fills every x-slot.
A structural ceiling, not an exact count.
Descending powers reveal degree quickly.
Many incorrect answers come from using a real algebra rule in the wrong context. Recognizing the source of an error is more useful than memorizing one correction.
x^2+x^3 does not become x^5. Exponents add when matching bases are multiplied.
4x^2+3x cannot be combined because the variable powers differ.
In -(x^2-4x+1), every sign changes: -x^2+4x-1.
The exponent gives degree; the number on the leading term is the leading coefficient.
Negative and multi-term inputs should be placed in parentheses before powers are applied.
The turning-point rule gives a maximum. A polynomial may have fewer turns.
These are illustrative teaching examples, not questions copied from the test.
(5x^4-3x^2+2)+(-2x^4+7x-6)
Result: 3x^4-3x^2+7x-4.
(2x^3+1)(x^5-4x)
Degrees 3 and 5 give product degree 8.
For P(x)=x^3-2x+4, P(-3)=-27+6+4=-17.
Dividing by x+1=x-(-1) means evaluate P(-1).
If P(4)=0, then x-4 is a factor and 4 is a real x-intercept.
A leading term -3x^6 has even degree and negative lead, so both ends fall.
Before choosing an answer, run a short structural check. It catches many mistakes faster than repeating the whole calculation.