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Quizzes > High School Quizzes > Mathematics

Polynomials Practice Quiz: Test Your Skills

Explore Graphs of Polynomial Functions with Confidence

Difficulty: Moderate
Grade: Grade 10
Study OutcomesCheat Sheet
Paper art promoting a high school practice quiz on polynomial concepts for exam preparation.

Which of the following expressions is a polynomial?
3x^2 + 2x - 5
x^-1 + 7
3/x + 2
√x + 4
A polynomial is composed of terms with non-negative integer exponents. The expression 3x^2 + 2x - 5 meets this definition.
What is the degree of the polynomial 4x^3 - 3x + 2?
1
4
3
2
The degree of a polynomial is determined by the highest exponent on x. In 4x^3 - 3x + 2, the highest exponent is 3.
In the polynomial 7x^4 - 2x^2 + x - 8, what is the constant term?
7
-8
x
-2x^2
The constant term is the term that does not contain any variables. In this polynomial, -8 is the constant term.
Which term in the polynomial 5x^3 + 4x^2 - 9x + 1 is the leading term?
4x^2
1
5x^3
-9x
The leading term is the term with the highest power of x. Here, 5x^3 has the highest exponent (3), so it is the leading term.
Which of the following standard forms represents a linear polynomial?
x^2 + 2x + 1
3x + 7
2x^2 - 3
4x^3 - x
A linear polynomial has a degree of 1. The expression 3x + 7 fits this criteria, while the other expressions have higher degrees.
What is the result of adding the polynomials 3x^2 + 2x - 5 and -x^2 + 4x + 3?
4x^2 + 2x - 2
2x^2 + 6x - 2
2x^2 - 2x - 8
4x^2 + 6x - 2
Combine like terms: (3x^2 - x^2) gives 2x^2, (2x + 4x) gives 6x, and (-5 + 3) gives -2. Thus, the sum is 2x^2 + 6x - 2.
What is the coefficient of the x term in the polynomial 6x^3 - 5x^2 + x - 4?
6
-4
-5
1
The coefficient of the x term is the number multiplying x in that term. Here, the term is simply x, which implies a coefficient of 1.
Which property justifies that 2(x + 3) is equivalent to 2x + 6 when expanding?
Identity property
Commutative property
Associative property
Distributive property
The Distributive property allows multiplication to be distributed over addition, which explains why 2(x + 3) expands to 2x + 6.
Simplify the expression x^2 - 4 using factorization.
(x - 4)(x + 0)
(x - 2)^2
(x + 4)(x - 1)
(x - 2)(x + 2)
The expression x^2 - 4 is a difference of squares and factors into (x - 2)(x + 2).
When multiplying the polynomials (x + 2)(x^2 - x + 3), what is the term with the highest degree?
x^3
x^4
3x
x^2
Multiplying the highest degree terms from each binomial gives x * x^2 = x^3, which is the term of the highest degree in the expanded form.
What is the product of the leading coefficients when multiplying the polynomials 2x^2 and 3x^3?
5
8
3
6
The leading coefficients are 2 and 3, and their product is 2 Ã - 3 = 6.
Which of the following is NOT a property of polynomials?
They can have negative exponents
Their exponents are non-negative integers
They have a finite number of terms
They obey the commutative property of addition
Polynomials are defined to have non-negative integer exponents. Therefore, stating that they can have negative exponents is incorrect.
What is the remainder when a polynomial f(x) is divided by (x - a)?
f(x) evaluated at x = a
f(x - a)
f(a)
a
According to the Remainder Theorem, when a polynomial f(x) is divided by (x - a), the remainder is f(a).
Which method is efficient for dividing polynomials when the divisor is of the form x - c?
Synthetic division
Long division
Factoring
Substitution
Synthetic division is a streamlined method for dividing polynomials by a binomial of the form x - c.
Identify the degree of the polynomial obtained by multiplying (2x + 3) and (x^2 - x + 4).
2
4
5
3
The degree of a product is the sum of the degrees of each factor. Here, (2x + 3) is degree 1 and (x^2 - x + 4) is degree 2, so the result is degree 3.
Factor completely: x^3 - 3x^2 - 4x + 12.
(x - 2)(x + 2)(x - 3)
(x + 2)(x - 2)(x + 3)
(x - 3)^2 (x + 4)
x(x - 3)(x + 4)
By grouping terms, x^3 - 3x^2 - 4x + 12 can be factored as (x - 3)(x^2 - 4), and further as (x - 3)(x - 2)(x + 2).
Given the polynomial f(x) = x^4 - 5x^2 + 4, how many distinct real roots does it have?
0
2
3
4
Substituting u = x^2 transforms the equation into u^2 - 5u + 4 = 0, which factors to (u - 1)(u - 4) = 0. Thus, x^2 = 1 or 4, giving roots x = ±1 and x = ±2, four distinct real roots.
If a polynomial f(x) is divisible by both (x - 1) and (x - 2), what must be true about f(x)?
f(1) = 1 and f(2) = 1
f(x) equals zero only when x = 1 or 2
f(x) has factors x + 1 and x + 2
f(1) = 0 and f(2) = 0
The Factor Theorem states that if (x - a) is a factor of f(x), then f(a) must be zero. Thus, f(1) and f(2) are both 0.
Which theorem states that every polynomial of degree n has exactly n roots, counting multiplicities, in the complex number system?
Rational Root Theorem
Remainder Theorem
Fundamental Theorem of Algebra
Factor Theorem
The Fundamental Theorem of Algebra guarantees that any non-zero polynomial of degree n has exactly n roots in the complex number system when counting multiplicities.
Solve for x: Find all roots of the polynomial 2x^2 - 3x - 2 = 0.
-2 and -1/2
2 and 1/2
2 and -1/2
-2 and 1/2
Using the quadratic formula on 2x^2 - 3x - 2 = 0, the solutions are x = (3 ± 5)/4, which simplifies to x = 2 and x = -1/2.
0
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Study Outcomes

  1. Identify the degree and leading coefficient of a polynomial.
  2. Simplify polynomial expressions by combining like terms.
  3. Factor polynomials using common factoring and special products.
  4. Analyze polynomial equations to determine their zeros.
  5. Apply the Remainder and Factor Theorems to verify polynomial factors.

Polynomials Quiz: Graphs & Practice Test Cheat Sheet

  1. Structure of Polynomials - Polynomials are playful sums of terms where each term has a variable raised to a whole-number power. The highest power tells you the "degree" and the leading coefficient is the number in front of that top term. Understanding this layout helps you tackle problems with confidence and clarity! Lumen Learning summary
  2. Closure Properties - Polynomials love teamwork: when you add, subtract, or multiply them, you always end up with another polynomial. This magical "closure" means you can mix and match polynomials without ever leaving the polynomial family. It's like having an all-access pass to polynomial land! Explore the standards
  3. Adding & Subtracting - To combine polynomials, simply group like terms (same variable and exponent) and add or subtract their coefficients. For example, (3x² + 2x) + (5x² - 4x) becomes 8x² - 2x. Mastering this keeps you fluent in polynomial arithmetic and speeds up problem solving. OpenStax guide
  4. Multiplying with FOIL - Use the distributive property to multiply any polynomials, and for two-term binomials, FOIL (First, Outer, Inner, Last) is your shortcut. For instance, (x + 3)(x + 2) unfolds to x² + 5x + 6. With practice, this becomes second-nature - no calculator required! Elementary Algebra tips
  5. Special Products - Recognize the patterns: perfect square trinomials (a + b)² = a² + 2ab + b² and the difference of squares (a + b)(a - b) = a² - b². These shortcuts save time and keep your work neat. Spotting them is like finding hidden treasures in equations! Symbolab glossary
  6. Factoring Techniques - Factor by pulling out the greatest common factor (GCF), grouping, or using patterns like difference of squares and perfect squares. For example, x² - 9 factors to (x + 3)(x - 3). Factoring is reverse multiplication - it turns big expressions into bite-size pieces. Deep dive on Symbolab
  7. Remainder Theorem - Divide any polynomial f(x) by (x - a) and the remainder will be f(a). If you get zero, you've discovered a factor! This theorem turns long division into a quick evaluation - plug and play! Core Standards details
  8. Rational Zero Theorem - Predict which rational numbers might solve f(x)=0 by checking factors of the constant term over factors of the leading coefficient. This roadmap narrows your search for roots, so you don't have to guess blindly. College Algebra resource
  9. Fundamental Theorem of Algebra - Every polynomial of degree n has exactly n roots (real or complex), counting multiplicity. This theorem assures you that solutions always exist and gives insight into the full picture of any polynomial function. SparkNotes overview
  10. Graphing Behavior - Sketch polynomials by analyzing end behavior (based on degree and leading coefficient), x‑intercepts (the roots), and turning points (local max/min). This visual approach turns abstract formulas into colorful curves you can interpret at a glance! Fiveable key concepts
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