Hello future math whiz! Getting ready for an exam is smart, and I'm here to help you nail those polynomial concepts. Let's get you squared away with a quick review and then a practice quiz!
Quick Study Guide
- Definition: A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
- Key Components:
- Term: A single number, variable, or product of numbers and variables (e.g., $3x^2$, $5y$, $-7$).
- Coefficient: The numerical factor of a term (e.g., in $4x^3$, $4$ is the coefficient).
- Variable: A symbol (usually a letter) representing an unknown value (e.g., $x$, $y$).
- Exponent: The power to which a variable is raised (must be a non-negative integer for polynomials, e.g., $0, 1, 2, 3...$).
- Constant: A term without a variable (e.g., $5$, $-12$).
- Standard Form: Write terms in descending order of their exponents (e.g., $ax^n + bx^{n-1} + ... + c$). The term with the highest exponent is called the leading term, and its coefficient is the leading coefficient.
- Degree of a Polynomial: The highest exponent of the variable in the polynomial.
- Degree 0: Constant (e.g., $7$)
- Degree 1: Linear (e.g., $2x + 1$)
- Degree 2: Quadratic (e.g., $3x^2 - 5x + 2$)
- Degree 3: Cubic (e.g., $x^3 + 2x^2 - x + 4$)
- Classifying by Number of Terms:
- Monomial: 1 term (e.g., $5x^3$)
- Binomial: 2 terms (e.g., $2x + 7$)
- Trinomial: 3 terms (e.g., $x^2 - 3x + 1$)
- Polynomials with 4 or more terms are generally just called "polynomials."
- What is NOT a Polynomial: Expressions with:
- Negative exponents (e.g., $x^{-2}$)
- Fractional exponents (e.g., $x^{\frac{1}{2}}$ or $\sqrt{x}$)
- Variables in the denominator (e.g., $\frac{1}{x}$)
- Variables under a radical sign (e.g., $\sqrt{x}$)
- Variables in the exponent (e.g., $2^x$)
Practice Quiz
-
Which of the following is a polynomial?
- $3x^{-2} + 5x$
- $\frac{2}{x} + 7$
- $4x^3 - 2x + 1$
- $5^x - 3$
-
What is the degree of the polynomial $7x - 4x^5 + 2x^2 - 1$?
- 1
- 2
- 3
- 5
-
Identify the leading coefficient of the polynomial $8 - 3x^2 + 5x^4 - x$.
- $8$
- $-3$
- $5$
- $-1$
-
Classify the expression $9x^2 - 6$ by its number of terms.
- Monomial
- Binomial
- Trinomial
- Quadratic
-
Which term correctly describes the polynomial $2x^3 + 5x - 1$ based on its degree?
- Linear
- Quadratic
- Cubic
- Constant
-
Which of the following expressions is NOT a polynomial?
- $6x^4 + 3x^2 - 9$
- $x^3 - \frac{x}{2} + 7$
- $4\sqrt{x} + 2x$
- $10y^5$
-
When written in standard form, what is the constant term of the polynomial $2x - 5 + x^2 - 3x$?
- $2x$
- $-5$
- $x^2$
- $-1$
Click to see Answers
- C. $4x^3 - 2x + 1$ (All exponents are non-negative integers.)
- D. $5$ (The highest exponent in the polynomial, when terms are ordered, is $5$ from $-4x^5$.)
- C. $5$ (In standard form, the polynomial is $5x^4 - 3x^2 - x + 8$. The leading term is $5x^4$, so the leading coefficient is $5$.)
- B. Binomial (It has two terms: $9x^2$ and $-6$.)
- C. Cubic (The highest exponent, or degree, is $3$.)
- C. $4\sqrt{x} + 2x$ (The term $4\sqrt{x}$ can be written as $4x^{\frac{1}{2}}$, which has a fractional exponent. Polynomials require whole number, non-negative exponents.)
- B. $-5$ (First, combine like terms: $x^2 + (2x - 3x) - 5 = x^2 - x - 5$. The term without a variable is $-5$.)