ACT Math: Quadratics and Polynomials Quiz Quiz

Sharpen your skills with this quiz focused on ACT Math quadratics and polynomials, covering factorization, roots, solving equations, graph features, and relationships among coefficients. Strengthen your understanding of key algebraic concepts found in the ACT Math section with practical problem-solving questions.

1. Factoring Quadratic Expressions

   What is the factored form of the quadratic expression x² - 5x + 6?

   1. (x - 6)(x + 1)
   2. (x - 2)(x - 3)
   3. (x - 1)(x - 6)
   4. (x + 2)(x + 3)

   Explanation: The expression x² - 5x + 6 factors into (x - 2)(x - 3) because -2 and -3 multiply to 6 and add to -5. (x + 2)(x + 3) yields a positive 5x, not -5x. (x - 6)(x + 1) multiplies to -6, not 6. (x - 1)(x - 6) multiplies to 6 but adds to -7x, not -5x. Careful factoring ensures correct identification of the answer.

2. Solving Quadratic Equations

   If x² + 4x + 4 = 0, which value of x satisfies the equation?

   1. -4
   2. -2
   3. 2
   4. 0

   Explanation: x² + 4x + 4 equals (x + 2)², so the equation becomes (x + 2)² = 0, giving x = -2. The option -4 would come from x² + 8x + 16 = 0. Option 2 is not a root in this equation, and 0 does not satisfy the original equation when substituted. Recognizing perfect square trinomials prevents errors with distractors.

3. Identifying Polynomial Degree

   What is the degree of the polynomial 4x³ - 7x² + 2x - 5?

   1. 4
   2. 1
   3. 2
   4. 3

   Explanation: The highest power of x in the polynomial is 3, making the degree 3. Option 2 corresponds to a quadratic term, not the whole polynomial. Option 4 is incorrect since there is no x⁴ term. Option 1 is the degree of the linear term, not the entire expression. Always look for the largest exponent to determine the degree.

4. Finding Zeros of a Polynomial

   Which of the following is a zero of the polynomial f(x) = x² - 9?

   1. 0
   2. -1
   3. 1
   4. 3

   Explanation: Setting x² - 9 = 0 gives x = 3 or x = -3; 3 is listed, so it is a correct zero of the polynomial. Option -1 and 1 give values of -8 for f(x), not zero. Option 0 yields f(0) = -9. Only option 3 (and also -3, if it had been listed) makes the polynomial equal zero.

5. Coefficient Relationships in Quadratic Equations

   For the quadratic equation x² + px + q = 0, which statement is true about the sum of its roots?

   1. The sum of the roots is q
   2. The sum of the roots is -q
   3. The sum of the roots is -p
   4. The sum of the roots is p

   Explanation: For ax² + bx + c = 0, the sum of the roots is given by -b/a; with a = 1, b = p, so the sum is -p. Option p is incorrect as it omits the negative. Option q and -q refer to the product, not the sum, of roots. Knowing Vieta's formulas helps avoid these common mistakes.