Question

Now for each pair of polynomials p(x) and q(x) given below, find p(x) + q(x) and p(x) − q(x). Also compute p(1) + q(1) and p(1) − q(1) for each.

(i)

(ii)

(iii)

(iv)

(v)

Answer 1

(i)

Given: p(x) = 2x² − 5x + 1, q(x) = x² + 3x + 2

p(x) + q(x) = (2x² − 5x + 1) + (x² + 3x + 2)

= 2x² − 5x + 1 + x² + 3x + 2

= (2 + 1)x² + (−5 + 3) x + (1 + 2)

= 3x² − 2x + 3

p(x) − q(x) = (2x² − 5x + 1) − (x² + 3x + 2)

= 2x² − 5x + 1 − x² − 3x − 2

= (2 − 1) x² − (5 + 3) x + (1 − 2)

= x² − 8x − 1

Now, p(1) + q(1) = 3(1)² − 2(1) + 3

= 3 − 2 + 3

= 6 − 2

= 4

p(1) − q(1) = (1)² − 8(1) − 1

= 1 − 8 − 1

= −8

(ii)

Given: p(x) = 2x² + x + 1, q(x) = x² + x + 2

p(x) + q(x) = (2x² + x + 1) + (x² + x + 2)

= 2x² + x + 1 + x² + x + 2

= (2 + 1)x² + (1 + 1)x + (1 + 2)

= 3x² + 2x + 3

p(x) − q(x) = (2x² + x + 1) − (x² + x + 2)

= 2x² + x + 1 − x² − x − 2

= (2 − 1)x² + (1 − 1)x + (1 − 2)

= x² − 1

Now, p(1) + q(1) = 3(1)² + 2(1) + 3

= 3 + 2 + 3

= 8

p(1) − q(1) = (1)² − 1

= 1 − 1

= 0

(iii)

Given: p(x) = 5x + 1, q(x) = x² + x + 2

p(x) + q(x) = (5x + 1) + (x² + x + 2)

= 5x + 1 + x² + x + 2

= x² + (5 + 1)x + (1 + 2)

= x² + 6x + 3

p(x) − q(x) = (5x + 1) − (x² + x + 2)

= 5x + 1 − x² − x − 2

= −x² + (5 − 1) x + (1 − 2)

= −x² + 4x − 1

Now, p(1) + q(1) = (1)² + 6(1) + 3

= 1 + 6 + 3

= 10

p(1) − q(1) = −(1)² + 4(1) − 1

= −1 + 4 − 1

= 4 − 2

= 2

(iv)

Given: p(x) = x² − 3x + 2, q(x) = x² + 3x − 2

p(x) + q(x) = (x² − 3x + 2) + (x² + 3x − 2)

= x² − 3x + 2 + x² + 3x − 2

= (1 + 1)x² + (−3 + 3)x + (2 − 2)

= 2x²

p(x) − q(x) = (x² − 3x + 2) − (x² + 3x − 2)

= x² − 3x + 2 − x² − 3 x + 2

= (1 − 1)x² − (3 + 3)x + (2 + 2)

= −6x + 4

Now, p(1) + q(1) = 2(1)² = 2

p(1) − q(1) = −6(1) + 4

= −6 + 4

= −2

(v)

Given: p(x) = 4x² − 2x + 3, q(x) = −4x² + 2x + 2

p(x) + q(x) = (4x² − 2x + 3) + (−4x² + 2x + 2)

= 4x² − 2x + 3 − 4x² + 2x + 2

= (4 − 4)x² + (−2 + 2)x + (3 + 2)

= 5

p(x) − q(x) = (4x² − 2x + 3) − (−4x² + 2x + 2)

= 4x² − 2x + 3 + 4x² − 2x − 2

= (4 + 4)x² − (2 + 2)x + (3 − 2)

= 8x² − 4x + 1

Now, p(1) + q(1) = 5

p(1) − q(1) = 8(1)² − 4(1) + 1

= 8 − 4 + 1

= 9 − 4

= 5