## FIND DEGREE OF SUM OF DIFFERENCE OF TWO POLYNOMIALS

Find Degree of Sum and Difference of Two Polynomials :

Here we are going to see, how to find degree of sum and difference of two polynomials.

If p and q are nonzero polynomials, then

deg(p + q) ≤ maximum{deg p, deg q}

and

deg(p − q) ≤ maximum{deg p, deg q}.

## Finding Degree of Sum and Difference of Two Polynomials - Examples

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− 7x+ x2 + 9x + 2 + 1

=  10x− 6x+ 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

=  − 8x- 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)  =  2x+ x+ 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 - 4x+ 21x + 4

After having gone through the stuff given above, we hope that the students would have understood "Find Degree of Sum and Difference of Two Polynomials".

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