Answer
(a) The degree of the product is determined by the sum of the degrees of the original polynomials.
(b) The degree of the sum is determined by the original polynomial with the greatest degree.
Work Step by Step
1. Make pairs of polynomials.
$$(2x + 3) \space and \space (x^2 - 5x )$$ $$(4) \space and \space (5x^3 + 4)$$ $$(9x^5) \space and \space (2x^2)$$
(a)
2. Find the product of each each pair.
$$(2x + 3) \times (x^2 - 5x) $$ $$ (2x)(x^2) + (2x)(-5x) + 3(x^2) + 3(-5x)$$ $$ 2x^3 - 10x^2 + 3x^2 -15x$$
The degrees of the original polynomials are, respectively, 1 and 2. And the degree of the product is 3.
$$(4) \times (5x^3 + 4) = 4(5x^3) + 4(4) = 20x^3 + 16$$
The degrees of the original polynomials are, respectively, 0 and 3. And the degree of the product is 3.
$$(9x^5) \times (2x^2) = 18x^7$$
The degrees of the original polynomials are, respectively, 5 and 2. And the degree of the product is 7.
As we can notice, the degree of the product is determined by the sum of the degrees of the original polynomials.
(b)
3. Find the sum of each pair.
$$(2x + 3) + (x^2 -5x) = x^2 + 2x - 5x + 3 = x^2 -3x + 3$$
The degrees of the original polynomials are, respectively, 1 and 2. And the degree of the sum is 2.
$$(4) + (5x^3 + 4) = 5x^3 + 4 +4 = 5x^3 + 8$$
The degrees of the original polynomials are, respectively, 0 and 3. And the degree of the sum is 3.
$$(9x^5) + (2x^2) = 9x^5 + 2x^2$$
The degrees of the original polynomials are, respectively, 5 and 2. And the degree of the sum is 5.
As we can notice, the degree of the sum is determined by the original polynomial with the greatest degree.
