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Quiz: Polynomial Expressions

Test your understanding of polynomial classification, operations, and special products with these questions.

1. What is the degree of the polynomial 5x³ - 2x² + 7x - 1?

  1. 1
  2. 2
  3. 3
  4. 5
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The correct answer is C. The degree of a polynomial is the highest power of the variable that appears in the expression. In 5x³ - 2x² + 7x - 1, the highest power is 3, so the degree is 3. This polynomial is called a cubic polynomial. Option A (1) is the degree of the linear term, option B (2) is the degree of the quadratic term, and option D (5) is the leading coefficient, not the degree.

Concept Tested: Degree of Polynomial

See: Degree of a Polynomial

2. Which polynomial is written in standard form?

  1. 5x - 2x³ + 3x⁴ - 7
  2. 3x⁴ - 2x³ + 5x - 7
  3. -7 + 5x - 2x³ + 3x⁴
  4. x - 7 + 3x⁴ - 2x³
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The correct answer is B. Standard form arranges terms in descending order of degree (highest power first). Option B is 3x⁴ - 2x³ + 5x - 7, with exponents going from 4 to 3 to 1 to 0. Options A, C, and D all contain the same terms but not in descending order of exponents, so they are not in standard form.

Concept Tested: Standard Form of Polynomial

See: Standard Form of a Polynomial

3. Add the polynomials: (3x² + 5x - 2) + (2x² - 3x + 7)

  1. 5x² + 2x + 5
  2. 5x² + 8x + 5
  3. x² + 2x + 9
  4. 5x⁴ + 2x + 5
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The correct answer is A. To add polynomials, combine like terms. Group terms with the same degree: (3x² + 2x²) + (5x - 3x) + (-2 + 7) = 5x² + 2x + 5. Option B incorrectly adds 5x and -3x as 8x instead of 2x. Option C incorrectly combines the x² terms. Option D incorrectly shows x⁴ instead of x².

Concept Tested: Adding Polynomials

See: Adding Polynomials

4. What is the result of (5x² - 3x + 4) - (2x² + x - 6)?

  1. 3x² - 2x - 2
  2. 3x² - 4x + 10
  3. 7x² - 2x - 2
  4. 3x² + 2x + 10
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The correct answer is B. To subtract polynomials, distribute the negative sign to every term in the second polynomial, then combine like terms: 5x² - 3x + 4 - 2x² - x + 6 = (5x² - 2x²) + (-3x - x) + (4 + 6) = 3x² - 4x + 10. Option A incorrectly handles the constant terms. Option C incorrectly adds instead of subtracts the x² terms. Option D has incorrect signs for the x term.

Concept Tested: Subtracting Polynomials

See: Subtracting Polynomials

5. Using the FOIL method, multiply (x + 4)(x + 5).

  1. x² + 20
  2. x² + 9x + 20
  3. x² + 5x + 20
  4. 2x + 20
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The correct answer is B. Using FOIL: First (x·x = x²), Outer (x·5 = 5x), Inner (4·x = 4x), Last (4·5 = 20). Combining: x² + 5x + 4x + 20 = x² + 9x + 20. Option A forgets the middle term entirely. Option C only includes one of the two middle terms (5x but not 4x). Option D treats this as simple addition rather than polynomial multiplication.

Concept Tested: FOIL Method

See: The FOIL Method

6. Simplify 3x(2x² - 5x + 4).

  1. 6x³ - 15x² + 12x
  2. 5x³ - 2x² + 7x
  3. 6x² - 15x + 12
  4. 6x³ - 5x + 4
Show Answer

The correct answer is A. Multiply the monomial 3x by each term in the polynomial: 3x·2x² = 6x³, 3x·(-5x) = -15x², and 3x·4 = 12x. The result is 6x³ - 15x² + 12x. Option B shows incorrect coefficients. Option C fails to add the exponents correctly (should be x³ and x², not x² and x). Option D only distributes 3x to some terms, not all.

Concept Tested: Multiplying Polynomials

See: Multiplying Polynomials

7. Which expression represents the difference of squares pattern?

  1. (x + 3)²
  2. (x + 5)(x - 5)
  3. x² + 2x + 1
  4. (x + 2)(x + 3)
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The correct answer is B. The difference of squares pattern is (a + b)(a - b) = a² - b². Option B, (x + 5)(x - 5), follows this pattern with a = x and b = 5, giving x² - 25. Option A is a perfect square (sum squared). Option C is already in expanded form. Option D is a general binomial product, not a difference of squares.

Concept Tested: Difference of Squares

See: Difference of Squares

8. Expand (x + 3)² using the perfect square trinomial pattern.

  1. x² + 9
  2. x² + 3x + 9
  3. x² + 6x + 9
  4. x² + 6x + 3
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The correct answer is C. Using the pattern (a + b)² = a² + 2ab + b², with a = x and b = 3: x² + 2(x)(3) + 3² = x² + 6x + 9. Option A forgets the middle term (2ab). Option B has the wrong middle term coefficient (should be 6x, not 3x). Option D has the wrong constant term (should be 9, not 3).

Concept Tested: Perfect Square Trinomial

See: Perfect Square Trinomials

9. What is the greatest common factor (GCF) of 6x³ + 9x² - 12x?

  1. 3
  2. x
  3. 3x
  4. 6x³
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The correct answer is C. To find the GCF: the GCF of the coefficients (6, 9, 12) is 3, and the lowest power of x that appears in all terms is x¹. Therefore, the GCF is 3x. Option A only includes the numerical GCF. Option B only includes the variable part. Option D is one of the original terms, not the greatest common factor.

Concept Tested: Greatest Common Factor

See: Greatest Common Factor

10. A rectangle has length (2x + 3) and width (x - 1). Which expression represents the perimeter?

  1. 3x + 2
  2. 6x + 4
  3. 2x² + x - 3
  4. 3x + 4
Show Answer

The correct answer is B. Perimeter = 2(length) + 2(width) = 2(2x + 3) + 2(x - 1) = 4x + 6 + 2x - 2 = 6x + 4. This requires distributing and combining like terms. Option A only calculates one length plus one width. Option C multiplies length times width (area, not perimeter). Option D has incorrect coefficient for the x term.

Concept Tested: Perimeter Problems

See: Perimeter Problems

Quiz Complete

Review the explanations above to reinforce your understanding of polynomial expressions. For additional practice, refer back to the chapter content.