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Question

For each pair of polynomials p(x) and q(x) given below find the degree of p(x) + q(x) and p(x)q(x).

(i)

(ii)

(iii)

(iv)

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Solution

(i)

Given: p(x) = x^{3} − 2x^{2} + 4x + 1 and q(x) = 5x^{2} + 3x + 2

We know that the degree of a polynomial is the largest exponent of the variable in the polynomial.

The largest exponent of x in p(x) = 3

∴ Degree of p(x) = 3

The largest exponent of x in q(x) = 2

∴ Degree of q(x) = 2

Between p(x) and q(x), p(x) has a higher degree. Therefore, the degree of {p(x) + q(x)} is equal to the degree of p(x).

∴ Degree of {p(x) + q(x)} = 3

We know that degree of p(x) q(x) = degree of p(x) + degree of q(x)

∴ Degree of p(x) q(x) = 3 + 2 = 5

(ii)

Given: p(x) = x^{3} − 2x^{2} + 4x + 1 and q(x) = 2x^{2} − 4x + 2

We know that the degree of a polynomial is the largest exponent of the variable in the polynomial.

The largest exponent of x in p(x) = 3

∴ Degree of p(x) = 3

The largest exponent of x in q(x) = 2

∴ Degree of q(x) = 2

Between p(x) and q(x), p(x) has a higher degree. Therefore, the degree of {p(x) + q(x)} is equal to the degree of p(x).

∴ Degree of {p(x) + q(x)} = 3

We know that degree of p(x) q(x) = degree of p(x) + degree of q(x)

∴ Degree of p(x) q(x) = 3 + 2 = 5

(iii)

Given: p(x) = x^{3} − 2x^{2} + 4x + 1 and q(x) = −x^{3} + 1

We know that the degree of a polynomial is the largest exponent of the variable in the polynomial.

The largest exponent of x in p(x) = 3

∴ Degree of p(x) = 3

The largest exponent of x in q(x) = 3

∴ Degree of q(x) = 3

It can be observed that both p(x) and q(x) have the same degree.

However, the coefficients of the variable having the largest exponent in both p(x) and q(x) are same in magnitude but opposite in sign. So, they will cancel out while the addition of the polynomials p(x) and q(x) is carried out.

Therefore, the degree of {p(x) + q(x)} is the next largest exponent of the variable in p(x) or q(x).

The next largest exponent is of x^{2}.

∴ Degree of {p(x) + q(x)} = 2

We know that degree of p(x) q(x) = degree of p(x) + degree of q(x)

∴ Degree of p(x) q(x) = 3 + 3 = 6

(iv)

Given: p(x) = x^{3} − 2x^{2} + 4x + 1 and q(x) = −x^{3} + 2x^{2} + 1

We know that the degree of a polynomial is the largest exponent of the variable in the polynomial.

The largest exponent of x in p(x) = 3

∴ Degree of p(x) = 3

The largest exponent of x in q(x) = 3

∴ Degree of q(x) = 3

It can be observed that both p(x) and q(x) have the same degree.

However, the coefficients of the variable having the largest exponent in both p(x) and q(x) are same in magnitude but opposite in sign. So, they will cancel out while the addition of the polynomials p(x) and q(x) is carried out.

Similarly, the coefficients of x^{2} will cancel out.

Therefore, the degree of {p(x) + q(x)} is the next largest exponent of the variable in p(x) or q(x).

The next largest exponent is of x.

∴ Degree of {p(x) + q(x)} = 1

We know that degree of p(x) q(x) = degree of p(x) + degree of q(x)

∴ Degree of p(x) q(x) = 3 + 3 = 6

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