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Quiz: Algebra Foundations

Test your understanding of variables, expressions, number systems, and factoring techniques with these review questions.

1. In the expression \(-4x^2 + 7x - 9\), what is the coefficient of the \(x^2\) term?

  1. \(4\)
  2. \(-4\)
  3. \(2\)
  4. \(-9\)
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The correct answer is B. The coefficient is the number that multiplies the variable, including its sign. In \(-4x^2\), the number multiplying \(x^2\) is \(-4\), so the coefficient is \(-4\). A common mistake is to drop the negative sign and answer \(4\). The value \(-9\) is the constant term (no variable), and \(2\) is the exponent, not the coefficient.

Concept Tested: Coefficient

2. Which of the following numbers is irrational?

  1. \(0.\overline{27}\)
  2. \(\frac{22}{7}\)
  3. \(\sqrt{2}\)
  4. \(-5\)
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The correct answer is C. An irrational number cannot be written as a fraction of two integers and has a non-terminating, non-repeating decimal expansion. \(\sqrt{2} \approx 1.41421356...\) fits this description. The repeating decimal \(0.\overline{27}\) and the fraction \(\frac{22}{7}\) are both rational (note: \(\frac{22}{7}\) is only an approximation of \(\pi\), not \(\pi\) itself). The integer \(-5\) is also rational since \(-5 = \frac{-5}{1}\).

Concept Tested: Irrational Numbers

3. Which symbol correctly represents the set of integers?

  1. \(\mathbb{R}\)
  2. \(\mathbb{Q}\)
  3. \(\mathbb{N}\)
  4. \(\mathbb{Z}\)
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The correct answer is D. The letter \(\mathbb{Z}\) (from the German Zahlen, meaning numbers) denotes integers: \(\{..., -2, -1, 0, 1, 2, ...\}\). \(\mathbb{R}\) denotes real numbers, \(\mathbb{Q}\) denotes rational numbers, and \(\mathbb{N}\) denotes natural (counting) numbers. The nesting relationship is \(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\).

Concept Tested: Integers

4. What is the result of expanding \((x - 3)(x + 7)\)?

  1. \(x^2 + 4x - 21\)
  2. \(x^2 - 4x - 21\)
  3. \(x^2 + 10x - 21\)
  4. \(x^2 - 21\)
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The correct answer is A. Using FOIL: First \(x \cdot x = x^2\), Outer \(x \cdot 7 = 7x\), Inner \(-3 \cdot x = -3x\), Last \(-3 \cdot 7 = -21\). Combining: \(x^2 + 7x - 3x - 21 = x^2 + 4x - 21\). A common mistake is forgetting to combine the middle terms (giving answer D) or reversing the sign of the middle term.

Concept Tested: Expanding Brackets

5. Which of the following is the correct factoring of \(x^2 - 49\)?

  1. \((x - 7)^2\)
  2. \((x + 7)(x - 7)\)
  3. \((x + 7)^2\)
  4. \((x - 49)(x + 1)\)
Show Answer

The correct answer is B. This is a difference of squares with \(a = x\) and \(b = 7\), since \(49 = 7^2\). Applying the pattern \(a^2 - b^2 = (a + b)(a - b)\), we get \((x + 7)(x - 7)\). Option A would expand to \(x^2 - 14x + 49\), not \(x^2 - 49\). The difference of squares pattern only works when you have two perfect squares separated by a minus sign.

Concept Tested: Difference of Squares

6. Simplify: \(6x^2 - 3x + 2x^2 + 8x - 5\).

  1. \(4x^2 + 11x - 5\)
  2. \(8x^2 - 11x + 5\)
  3. \(8x^2 + 5x - 5\)
  4. \(13x^2 - 5\)
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The correct answer is C. Group like terms by matching variable and power: \((6x^2 + 2x^2) + (-3x + 8x) + (-5) = 8x^2 + 5x - 5\). Remember that only terms with the same variable raised to the same power can be combined. You cannot combine \(x^2\) terms with \(x\) terms, which is why option D is incorrect.

Concept Tested: Collecting Like Terms

7. What is the common factor of \(15a^3b - 10a^2b^2 + 25a^2b\)?

  1. \(5ab\)
  2. \(5a^2b\)
  3. \(15a^2b^2\)
  4. \(a^2b\)
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The correct answer is B. To find the greatest common factor, take the largest number that divides all coefficients (\(\gcd(15, 10, 25) = 5\)) and the smallest power of each shared variable (\(a^2\) and \(b^1\)). This gives \(5a^2b\). Factoring out yields \(5a^2b(3a - 2b + 5)\). Option A misses one factor of \(a\), and option D ignores the numerical coefficient entirely.

Concept Tested: Common Factor

8. Which expression is a perfect square trinomial?

  1. \(x^2 + 5x + 4\)
  2. \(x^2 + 4x + 16\)
  3. \(x^2 - 9\)
  4. \(x^2 - 6x + 9\)
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The correct answer is D. A perfect square trinomial fits the pattern \(a^2 \pm 2ab + b^2 = (a \pm b)^2\). For \(x^2 - 6x + 9\): the first term is \(x^2\), the last term is \(3^2\), and the middle term is \(2(x)(3) = 6x\) (with a negative sign). So \(x^2 - 6x + 9 = (x - 3)^2\). Option B fails because \(2(x)(4) = 8x\), not \(4x\).

Concept Tested: Perfect Square Trinomial

9. Solve the system \(\begin{cases} y = x + 4 \\ 2x + y = 13 \end{cases}\) using substitution.

  1. \(x = 3\), \(y = 7\)
  2. \(x = 4\), \(y = 8\)
  3. \(x = 5\), \(y = 9\)
  4. \(x = 2\), \(y = 6\)
Show Answer

The correct answer is A. Substitute \(y = x + 4\) into the second equation: \(2x + (x + 4) = 13\), which gives \(3x + 4 = 13\), so \(3x = 9\) and \(x = 3\). Then \(y = 3 + 4 = 7\). Check: \(2(3) + 7 = 13\). The substitution method is efficient here because one equation is already solved for \(y\).

Concept Tested: Substitution Method

10. Factor completely: \(x^3 + 27\).

  1. \((x + 3)(x^2 - 3x + 9)\)
  2. \((x + 3)^3\)
  3. \((x - 3)(x^2 + 3x + 9)\)
  4. \((x + 3)(x^2 + 3x + 9)\)
Show Answer

The correct answer is A. This is a sum of cubes with \(a = x\) and \(b = 3\), since \(27 = 3^3\). The formula is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Applying SOAP (Same, Opposite, Always Positive): the first bracket uses the same sign (\(+\)), the second bracket has the opposite sign on the \(ab\) term (\(-3x\)), and the last term is always positive (\(+9\)). Result: \((x + 3)(x^2 - 3x + 9)\).

Concept Tested: Sum and Difference Cubes