Question

which is true about the degree of the sum and difference of the polynomials \\(3x^5y - 2x^3y^4 - 7xy^3\\) and \\(-8x^5y + 2x^3y^4 + xy^3\\)?

both the sum and difference have a degree of 6.
both the sum and difference have a degree of 7.
the sum has a degree of 6, but the difference has a degree of 7.
the sum has a degree of 7, but the difference has a degree of 6.

Explanation:

Identify the given polynomials

Let the two polynomials be:

$$ P(x, y) = 3x^5y - 2x^3y^4 - 7xy^3 $$

$$ Q(x, y) = -8x^5y + 2x^3y^4 + xy^3 $$

Calculate the sum of the polynomials

Combine like terms:

$$ P(x, y) + Q(x, y) = (3x^5y - 8x^5y) + (-2x^3y^4 + 2x^3y^4) + (-7xy^3 + xy^3) $$

$$ P(x, y) + Q(x, y) = -5x^5y - 6xy^3 $$

Determine the degree of the sum

Find the degree of each term in the sum:

• For \(-5x^5y\), the degree is \(5 + 1 = 6\).
• For \(-6xy^3\), the degree is \(1 + 3 = 4\).

The degree of the sum is the highest term degree:

$$ \text{Degree of Sum} = 6 $$

Calculate the difference of the polynomials

Subtract \(Q(x, y)\) from \(P(x, y)\):

$$ P(x, y) - Q(x, y) = (3x^5y - (-8x^5y)) + (-2x^3y^4 - 2x^3y^4) + (-7xy^3 - xy^3) $$

$$ P(x, y) - Q(x, y) = 11x^5y - 4x^3y^4 - 8xy^3 $$

Determine the degree of the difference

Find the degree of each term in the difference:

• For \(11x^5y\), the degree is \(5 + 1 = 6\).
• For \(-4x^3y^4\), the degree is \(3 + 4 = 7\).
• For \(-8xy^3\), the degree is \(1 + 3 = 4\).

The degree of the difference is the highest term degree:

$$ \text{Degree of Difference} = 7 $$

Answer:

• Both the sum and difference have a degree of 6.
• Both the sum and difference have a degree of 7.
• The sum has a degree of 6, but the difference has a degree of 7. (Correct answer)
• The sum has a degree of 7, but the difference has a degree of 6.