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Quizzes > High School Quizzes > Mathematics

Unit 7 Polynomials Practice Quiz

Sharpen Your Skills with Review Questions and Answers

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Paper art representing a Polynomial Power-Up quiz for high school algebra students.

What is the degree of the polynomial 5x^3 + 2x^2 - 7?
2
0
7
3
The degree of a polynomial is determined by the highest exponent of the variable. In the expression 5x^3 + 2x^2 - 7, the highest power of x is 3, so the degree is 3.
Which term is the constant term in the polynomial 4x^2 - 3x + 8?
-3x
4x^2
x^2
8
The constant term is the term without any variable attached. In the polynomial 4x^2 - 3x + 8, only 8 has no x, making it the constant term.
What is the sum of the polynomials (2x + 3) and (4x - 5)?
6x + 2
2x - 8
6x - 2
8x - 2
To add the two polynomials, combine like terms: add the x-terms (2x + 4x) to get 6x and add the constants (3 - 5) to get -2. The sum is therefore 6x - 2.
Which polynomial is written in standard form?
3 + x^2
x + 3
4x + 5x^2
x^2 + 4x + 4
Standard form of a polynomial requires the terms to be written in descending order of degree. The polynomial x^2 + 4x + 4 arranges its terms from the highest degree to the lowest.
What is the product of 2x and 3x^2?
5x^3
6x^2
6x^3
5x^2
Multiply the coefficients (2 and 3) to get 6 and add the exponents on x (1 and 2) to get x^(1+2) or x^3. Therefore, the product is 6x^3.
Factor the expression 6x^3 + 9x^2 completely.
9x(2x + 1)
x^2(6x + 9)
3x(2x + 3)
3x^2(2x + 3)
The greatest common factor of 6x^3 and 9x^2 is 3x^2. Factoring out 3x^2 leaves the binomial (2x + 3), which is the complete factorization.
What is the product of (x + 2) and (x - 3)?
x^2 + 6
x^2 + x - 6
x^2 - x - 6
x^2 - 5
Using the FOIL method, multiply the binomials: x*x gives x^2, the outer and inner products give -3x and +2x respectively (which combine to -x), and 2*(-3) gives -6. This results in x^2 - x - 6.
Subtract the polynomial (3x^3 + x^2) from (5x^3 - 2x^2) to find the result.
2x^3 - x^2
8x^3 - 3x^2
2x^3 + 3x^2
2x^3 - 3x^2
Subtract corresponding like terms: 5x^3 minus 3x^3 equals 2x^3, and -2x^2 minus x^2 equals -3x^2. Thus, the resulting polynomial is 2x^3 - 3x^2.
What is the remainder when x^2 + 3x + 2 is divided by (x + 1)?
0
-2
2
1
Using the Remainder Theorem, you substitute x = -1 into the polynomial: (-1)^2 + 3(-1) + 2 equals 1 - 3 + 2, which simplifies to 0. Therefore, the remainder is 0.
Using the Factor Theorem, if f(-2) = 0 for the polynomial f(x) = x^3 + 4x^2 + x - 6, which factor can be deduced?
(x + 3)
(x + 2)
(x - 2)
(x - 3)
The Factor Theorem states that if f(c) = 0, then (x - c) is a factor of f(x). Here, since f(-2) = 0, the factor is (x + 2).
Find the missing coefficient k such that x^2 + kx + 9 forms a perfect square trinomial.
-6
6
3
9
A perfect square trinomial has the form (x + a)^2 = x^2 + 2ax + a^2. Since 9 is a perfect square (3^2), a must be 3, so k is 2a = 6.
Simplify the expression: (7x^2 + 3x) - (2x - 5).
7x^2 + 2x - 5
5x^2 + x + 7
7x^2 + 5x - 5
7x^2 + x + 5
First, remove the parentheses and change the sign of each term in the second group, then combine like terms. This gives 7x^2 + (3x - 2x) + 5 = 7x^2 + x + 5.
If f(x) = 2x^2 - 5x + 3, what is the value of f(2)?
-1
2
1
3
Plug x = 2 into the function: f(2) = 2(2)^2 - 5(2) + 3 = 8 - 10 + 3, which simplifies to 1.
Which of the following polynomials is cubic with a leading coefficient of 1?
3x^4 + x^3 - 1
x^3 + 2x^2 - x + 7
2x^3 + x^2 - x + 7
x^2 + 3x + 4
A cubic polynomial has a degree of 3. The polynomial x^3 + 2x^2 - x + 7 satisfies this condition and has a leading coefficient of 1.
Simplify the expression: 3x^2 - 2x + 5 - (x^2 + x - 4).
x^2 - 3x + 9
2x^2 + 3x + 1
4x^2 - x + 1
2x^2 - 3x + 9
Begin by distributing the negative sign to the terms inside the second parentheses, then combine like terms: (3x^2 - x^2) gives 2x^2, (-2x - x) gives -3x, and (5 + 4) gives 9.
Perform the polynomial long division: Divide 2x^3 + 3x^2 - x + 5 by x + 2.
2x^2 - x + 1 with a remainder of 3
2x^2 - x + 1 with no remainder
2x^2 + x - 1 with a remainder of 3
2x^2 + 3x + 5
Using long division, divide the leading term 2x^3 by x to get 2x^2, and continue the process to obtain a quotient of 2x^2 - x + 1 with a remainder of 3.
Using the Rational Root Theorem, identify one possible rational zero of f(x) = x^3 - 4x^2 + x + 6.
3
2
-3
-2
The Rational Root Theorem tells us that any rational zero is a factor of the constant term over a factor of the leading coefficient. Testing these possibilities, x = 2 yields f(2) = 0, so 2 is a zero.
Which polynomial has end behavior that approaches infinity as x approaches both positive and negative infinity?
x^4 - 5x^2 + 3
-x^2 + 4
x^3 - 4x
-x^4 + 2x^3
A polynomial's end behavior is determined by its degree and the sign of its leading coefficient. An even-degree polynomial with a positive leading coefficient, like x^4 - 5x^2 + 3, will approach infinity as x approaches both positive and negative infinity.
Factor the polynomial completely: x^3 - 6x^2 + 11x - 6.
(x - 6)(x^2 + 11)
(x - 1)(x - 2)(x - 3)
x(x - 2)(x - 3)
(x - 1)^3
Testing possible roots reveals that x = 1, 2, and 3 are zeros of the polynomial. Consequently, the complete factorization is (x - 1)(x - 2)(x - 3).
Determine the zeros of the polynomial 2x^3 - 3x^2 - 8x + 12.
x = -2, 3/2, 2
x = -2, 2, 3
x = 2, 3, -1
x = -3/2, 2, -2
Factoring by grouping yields (2x - 3)(x^2 - 4) = (2x - 3)(x - 2)(x + 2). Setting each factor equal to zero gives the zeros: x = 3/2, 2, and -2.
0
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Study Outcomes

  1. Analyze the structure of polynomials by identifying terms, coefficients, and degrees.
  2. Apply operations such as addition, subtraction, multiplication, and division to polynomial expressions.
  3. Simplify and factor polynomial expressions using appropriate factoring techniques.
  4. Solve polynomial equations and inequalities to determine their solutions.
  5. Evaluate problem-solving strategies to identify errors and improve understanding of polynomial concepts.

Unit 7 Polynomials Review Cheat Sheet

  1. Degree of a Polynomial - The degree is basically the "biggest" exponent in your polynomial, like the top scorer in a game! Knowing the degree helps you predict behavior and graph shapes. It's the foundation for everything else in polynomial adventures. GeeksforGeeks: Polynomials
  2. Classify Polynomials - Learn your monomials, binomials, and trinomials by counting terms like you'd count slices of pizza - one, two, or three! This simple taxonomy helps you choose the right strategies for solving and factoring. Get comfortable with these labels to speed up your problem solving. GeeksforGeeks: Polynomials
  3. Standard Form - Writing a polynomial in standard form means arranging terms from highest exponent to lowest, like lining up your team from tallest to shortest. This neat order makes adding, subtracting, and graphing a breeze. It's your go-to format for clarity and confidence. GeeksforGeeks: Polynomials
  4. Polynomial Identities - Memorize classics like (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b² - they're your magic wands for quick expansions. These identities slash through complex expressions in one swift move, saving time and brainpower. Practice them until you can recite them in your sleep! BYJU's: Polynomial Identities
  5. Factoring Polynomials - Break down polynomials into products of simpler expressions, just like dismantling LEGO builds block by block. Spot common factors, use grouping, or apply formulas to crack x² + 5x + 6 into (x + 2)(x + 3). Factoring is the key to solving equations and simplifying expressions. GeeksforGeeks: Factoring Methods
  6. Vieta's Formulas - Turn the coefficients of ax² + bx + c = 0 into treasure maps for roots: the sum is - b/a and the product is c/a. These shortcuts are perfect for checking your factorization and finding relationships between roots, even when the numbers get messy. Wikipedia: Vieta's Formulas
  7. Remainder Theorem - When you divide P(x) by x - a, the remainder is simply P(a). No long division needed! This trick lets you evaluate polynomials rapidly and check for factors in a snap. Think of it as your express lane through polynomial traffic. BYJU's: Remainder Theorem
  8. Factor Theorem - If P(a) = 0, then x - a is a factor of P(x). This principle transforms root-finding into a straightforward test: plug in candidate values, and when the result is zero, you've scored a perfect factor. It's detective work for polynomial puzzles! BYJU's: Factor Theorem
  9. Polynomial Division - Master long division and synthetic division to split polynomials and uncover quotients and remainders. Think of it like traditional division but with variables - once you nail the steps, you'll handle even high-degree polynomials with ease. SparkNotes: Polynomial Division
  10. Intermediate Value Theorem - For any continuous polynomial P(x), if P(a) and P(b) have opposite signs, there's at least one root between a and b. This theorem is your guarantee that you won't miss any hidden zeros when sketching graphs or estimating solutions. BYJU's: Intermediate Value Theorem
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