FIND DEGREE OF SUM OF DIFFERENCE OF TWO POLYNOMIALS

Question 1 :

Suppose p and q are polynomials defined by

p(x) = 2 − 7x² + 5x³ and q(x) = 1 + 9x + x² + 5x³

(i)  What is deg (p + q) (x)

(ii)  What is deg (p - q) (x)

Solution :

p(x) = 2 − 7x² + 5x³ and q(x) = 1 + 9x + x² + 5x³

(i)  

 (p + q) (x)  =  2 − 7x² + 5x³ + 1 + 9x + x² + 5x³

  =  5x³ + 5x³− 7x² + x² + 9x + 2 + 1

  =  10x³ − 6x² + 9x + 3

The highest power of the sum of two polynomials is 3. Hence the degree of (p + q) (x) is 3.

(ii)  

 (p - q) (x)  =  2 − 7x² + 5x³ - (1 + 9x + x² + 5x³)

  =  2 − 7x² + 5x³ - 1 - 9x - x² - 5x³

  =  − 7x² - x² - 9x + 2 - 1

  =  − 8x² - 9x + 1

The highest power of the sum of two polynomials is 2. Hence the degree of (p - q) (x) is 2.

Question 2 :

Write the indicated expression as a sum of terms, each of which is a constant times a power of x.

(i)  p(x) = x² + 5x + 2, q(x) = 2x³ − 3x + 1

Solution :

p(x) = x² + 5x + 2, q(x) = 2x³ − 3x + 1

(p + q) (x)  =  x² + 5x + 2 + 2x³ − 3x + 1

  =  2x³ + x² + 5x - 3x + 2 + 1

(p + q) (x)  =  2x³ + x² + 2x + 3

(ii)  (3p − 2q)(x)

(3p − 2q)(x) = 3(x² + 5x + 2) - 2(2x³ − 3x + 1)

(3p − 2q)(x) = 3x² + 15x + 6 - 4x³ + 6x - 2

=  - 4x³ + 3x²  + 15x + 6x + 6 - 2

= 3x² - 4x³ + 21x + 4

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