INTRODUCTORY EXERCISES - Polynomials and Roots

---

(1) If p(x) = 4 - 3x + ¹/₂x², then find:

    (a) p(1)
    (b) p(1/3)
    (c) p(-)

( Click On SOLUTION To See The Answer )

(2) Solve for the indicated variable.

    (a) x2 + 2xy - 12 = 0, y
    (b) 2r2 + 2rh = 2500, h

( SOLUTION )

(3) Solve each of the following quadratic equations by factoring.

    (a) -11q + 2q2 + 5 = 0
    (b) 4u2 = 8u
   

( SOLUTION )

(4) Solve the following quadratic equations by using the square root property. Example: x² = 25 => x = 5 or x = -5.

    (a) 9y2 - 16 = 0
    (b) (d - 3)2 = 3/4
    (c) 16x2 + 9 = 34

( SOLUTION )

(5) Solve the following quadratic equation by completing the square.

    x2 + 10x - 2 = 0

( SOLUTION )

(6) Find all real solution(s) for the following equations by using the quadratic formula. (Note: some equations may not have any real solutions.)

     (a) 3a(a-5) = 4a2 + 10a - 20
     (b) 13x2 +17x + 21 = 4x2 + 5x + 17

( HINT , SOLUTION )

(7) Given the following fraction, divide the denominator into the numerator using long division. Give the quotient and the remainder.

(a) 7y2 + 2y - 63
    _____________  
        y + 3

(b)  6x3 - 5x2 + 15x + 5
    ---------------------
        2x2 - x + 3

(c)      x6 + a6
    ----------------
     x4  - a2x2 + a4

( SOLUTION )

(8) Use the fact about roots of polynomials (the Factor Theorem) to find all real roots of the following polynomial equation.

    a3 - 6 = 6a2 - 11a

( SOLUTION )

---