Question

1. if a = xy³ + 2y - 3 and b = -3xy³ + 3y - 3x⁵, find each of the following:

a) the result of a − b: (2 marks)
(xy³ + 2y - 3) + 3xy - 3y + 3x⁵
xy³ + 2y - 3 + 3xy -
xy³ - y - 3 + 3xy

b) the degree of each polynomial: (2 marks)
a has degree 4
b has degree 5

1. construct a trinomial with a degree of 6, using only the variables x and y. (3 marks)

x⁶ + y + y
like terms

1. explain the difference between a constant and a numerical coefficient. (2 marks)

support your answer with examples.
constant: a number by itse
ex 3y - 2, ½, 3
numerical coefficient: the number
ex. ③xy, ②a²bc, ①abc

Explanation:

4a) Solving \( A - B \)

Step 1: Substitute \( A \) and \( B \)

Given \( A = xy^3 + 2y - 3 \) and \( B = -3xy^3 + 3y - 3x^5 \), so \( A - B=(xy^3 + 2y - 3)-(-3xy^3 + 3y - 3x^5) \)

Step 2: Distribute the negative sign

\( = xy^3 + 2y - 3 + 3xy^3 - 3y + 3x^5 \)

Step 3: Combine like terms

For \( xy^3 \) terms: \( xy^3+3xy^3 = 4xy^3 \)
For \( y \) terms: \( 2y - 3y=-y \)
Constant term: \( - 3\)
Remaining term: \( 3x^5 \)
So \( A - B = 3x^5+4xy^3 - y - 3 \)

Step 1: Degree of \( A \)

For polynomial \( A=xy^3 + 2y - 3 \), the degree of a term is the sum of exponents of variables. For \( xy^3 \), degree is \( 1 + 3=4 \), for \( 2y \) degree is \( 1 \), for \( - 3 \) degree is \( 0 \). The highest degree term is \( xy^3 \) with degree \( 4 \), so degree of \( A \) is \( 4 \)

Step 2: Degree of \( B \)

For polynomial \( B=-3xy^3 + 3y - 3x^5 \), for \( -3xy^3 \) degree is \( 1+3 = 4 \), for \( 3y \) degree is \( 1 \), for \( -3x^5 \) degree is \( 5 \). The highest degree term is \( -3x^5 \) with degree \( 5 \), so degree of \( B \) is \( 5 \)

Step 1: Recall trinomial and degree

A trinomial has 3 terms. The degree of a term with variables \( x \) and \( y \) is \( \text{degree of }x+\text{degree of }y \). We need the highest degree among terms to be \( 6 \)

Step 2: Create terms

Let's create three terms. For example, take \( x^6 \) (degree \( 6 \)), \( x^2y^4 \) (degree \( 2 + 4=6 \)), and \( y^6 \) (degree \( 6 \)) is not good as we need three distinct terms (or not necessarily distinct but sum to degree 6). A simple trinomial: \( x^6+xy^5 + y \) (degree of \( x^6 \) is \( 6 \), degree of \( xy^5 \) is \( 1+5 = 6 \), degree of \( y \) is \( 1 \), but the highest degree is \( 6 \)) or \( x^3y^3+x^2y^4 + x^6 \)
A valid trinomial: \( x^6+xy^5 + y^2 \) (degree of \( x^6 \) is \( 6 \), degree of \( xy^5 \) is \( 6 \), degree of \( y^2 \) is \( 2 \), highest degree is \( 6 \))

Answer:

\( 3x^5 + 4xy^3 - y - 3 \)

4b) Degree of Polynomials