Question

Find polynomials p(x) satisfying each set of conditions below.

i. First degree polynomials with p(1) = 1 and p(2) = 3

ii. First degree polynomials with p(1) = -1 and p(-2) = 3

iii. Second degree polynomials with p(0) = 0, p(1) = 2 and p(2) = 6.

iv. Three different second degree polynomials with p(0) = 0 and p(1) = 2.

Answer 1

i. General form of a first degree polynomial is

p(x) = ax + b

Let p(1) = 1, then a × 1 + b = l

a + b = 1 ………. (1)

Let p(2) = 3, then a × 2 + b = 3

2a + b = 3 …….. (2)

(1) × 2, 2a + 2b = 2 …… (3)

(3) – (2), b = -1

From (1), a + -1 = 1,

a = 1 + 1 = 2 Polynomial p(x) = 2x – 1

ii. General form of a first degree polynomial is p(x) = ax + b

Let p(1) = -1, then a × 1 + b = -l

a + b = -1 ……….. (1)

Let p(-2) = 3 , then a × (-2) + b = 3

-2a + b = 3 ………. (2)

(1) × 2, 2a + 2b = -2 ………. (3)

(2) + (3), 3b = 1, b = 1/3

From (1), a = 1/3 = -1

iii. General form of a second degree polynomial is

p(x) = ax² + bx + c

Let p(0) = 0, then a × 0² + b × 0 + c = 0

0a + 0b + c = 0.

c = 0 (1)

Let p(1) = 2 ,then a × 1² + b × 1 + c = 2

a + b + 0 = 2

a + b = 2 ………. (2)

Let p(2)= 6, then a × 2² + b × 2 + c = 6

4a + 2b = 6 

2a + b = 3 (3) 

(3) – (2), a = 1

From (2), 1 + b = 2, b = 2 – 1 = 1 

Polynomial p(x) = x² + x

iv. General form of a second degree 

polynomial is p(x) = ax² + bx + c 

Let p(0) = 0, then a x 0 + b x 0 + c = 0 

0 + 0 + c = 0 

c = 0 

Let p(1) = 2, then a × 1² + b × 1 + c = 2 

a + b + c = 2 

a + b = 2 

Selecting a and b such that a + b = 2 will give different polynomials. 

a = 1, b = 1 

a= 3, b = -l 

a = 4, b = -2 

Three different second degree polynomials are 

p(x) = x² + x 

p(x) = 3x² – x 

p(x) = 4x² – 2x