**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}.

**Question 1 :**

Suppose p and q are polynomials defined by

p(x) = 2 − 7x^{2} + 5x^{3} and q(x) = 1 + 9x + x^{2} + 5x^{3}

(i) What is deg (p + q) (x)

(ii) What is deg (p - q) (x)

**Solution :**

p(x) = 2 − 7x^{2} + 5x^{3} and q(x) = 1 + 9x + x^{2} + 5x^{3}

(i)

(p + q) (x) = 2 − 7x^{2} + 5x^{3} + 1 + 9x + x^{2} + 5x^{3}

= 5x^{3 }+ 5x^{3}− 7x^{2 }+ x^{2 }+ 9x + 2 + 1

= 10x^{3 }− 6x^{2 }+ 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 − 7x^{2} + 5x^{3} - (1 + 9x + x^{2} + 5x^{3})

= 2 − 7x^{2} + 5x^{3} - 1 - 9x - x^{2} - 5x^{3}

= − 7x^{2} - x^{2} - 9x + 2 - 1

= − 8x^{2 }- 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) = x^{2} + 5x + 2, q(x) = 2x^{3} − 3x + 1

**Solution :**

p(x) = x^{2} + 5x + 2, q(x) = 2x^{3} − 3x + 1

(p + q) (x) = x^{2} + 5x + 2 + 2x^{3} − 3x + 1

= 2x^{3 }+ x^{2 }+ 5x - 3x + 2 + 1

(p + q) (x) = 2x^{3 }+ x^{2 }+ 2x + 3

(ii) (3p − 2q)(x)

(3p − 2q)(x) = 3(x^{2} + 5x + 2) - 2(2x^{3} − 3x + 1)

(3p − 2q)(x) = 3x^{2} + 15x + 6 - 4x^{3} + 6x - 2

= - 4x^{3 }+ 3x^{2 }+ 15x + 6x + 6 - 2

= 3x^{2} - 4x^{3 }+ 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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