We will call this sentence a division statement. Activity 1 Perform the following operations. For each operation identify the dividend, the divisor, quotient and remainder. b) c) Using the method outline, lets extend our method outlined above to polynomials. Examples 1. Find the quotient and the remainder We begin by writing + 4x 5 in x2 – when the polynomial + 4x 5 is terms of a quotient and a remainder x2 – divided by gives an expression in the form: + 3 + 4x 5 (x +3).Q(x) + Remainder x x2 – We observe that the quotient must be of the form: + So, ax b. + 4x 5 (x +3)(ax + + R x2 b) – + + 3ax + 3b + R ax2 bx + (b + 3a)x +(3b +R) ax2 Comparing coefficients: : = 1………..(1) x2 a + 3a = 4…….(2) x: b + 3(1) = 4 b = 1 b : 3b + R = x0 –5 3(1) + R = 5 – R = 8 – Q(x) = (x + 1) ; R = 8 – 2. (x +2).Q(x) + R We observe that the quotient must be a quadratic, i.e: Q(x) = + + c ax2 bx (x +2)( + + c) + R ax2 bx + (2a + + (2b + + (2c + ax3 b)x2 c)x R) Comparing coefficients: = 2……..(1) x3: a
We will call this sentence a division statement. Activity 1 Perform the following operations. For each operation identify the dividend, the divisor, quotient and remainder. b) c) Using the method outline, lets extend our method outlined above to polynomials. Examples 1. Find the quotient and the remainder when the polynomial x2 + 4x – 5 is divided by x + 3 We begin by writing x2 + 4x – 5 in terms of a quotient and a remainder gives an expression in the form: x2 + 4x – 5 (x +3).Q(x) + Remainder We observe that the quotient must be of the form: ax + b. So, x2 + 4x – 5 (x +3)(ax + b) + R ax2 + bx + 3ax + 3b + R ax2 + (b + 3a)x +(3b +R) Comparing coefficients: x2 : a = 1………..(1) x: b + 3a = 4…….(2) b + 3(1) = 4 b = 1 x0 : 3b + R = –5 3(1) + R = – 5 R = – 8 Q(x) = (x + 1) ; R = – 8 2. (x +2).Q(x) + R We observe that the quotient must be a quadratic, i.e: Q(x) = ax2 + bx + c (x +2)( ax2 + bx + c) + R ax3 + (2a + b)x2 + (2b + c)x + (2c + R) Comparing coefficients: x3: a = 2……..(1)
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