Question

topic 1: polynomial basics

1. write in standard form the following

polynomials in standard form:
a) $5a + 3 - 4a^2 + 2a^3$

b) $7xy + x^3 - 3y^3 - 5x^2y^2$

a.)
b.)

Explanation:

Step1: Recall Standard Form of Polynomial

The standard form of a polynomial is written by arranging the terms in descending order of their degrees (exponents of the variable).

Step2: Solve Part (a)

For the polynomial \(5a + 3 - 4a^{2}+2a^{3}\), we identify the degree of each term:

• Degree of \(2a^{3}\) is \(3\)
• Degree of \(- 4a^{2}\) is \(2\)
• Degree of \(5a\) is \(1\)
• Degree of \(3\) (constant term) is \(0\)

Now, arrange the terms in descending order of their degrees:
\(2a^{3}-4a^{2}+5a + 3\)

Step3: Solve Part (b)

For the polynomial \(7xy+x^{3}-3y^{3}-5x^{2}y^{2}\), we first find the degree of each term. The degree of a term with multiple variables is the sum of the exponents of the variables:

• Degree of \(x^{3}\): \(3\) (since exponent of \(x\) is \(3\), \(y\) is \(0\))
• Degree of \(-5x^{2}y^{2}\): \(2 + 2=4\) (sum of exponents of \(x\) and \(y\))
• Degree of \(7xy\): \(1+1 = 2\)
• Degree of \(-3y^{3}\): \(3\) (exponent of \(y\) is \(3\), \(x\) is \(0\))

Now, arrange the terms in descending order of their degrees:
\(-5x^{2}y^{2}+x^{3}-3y^{3}+7xy\) (since degree \(4\) term comes first, then degree \(3\) terms, then degree \(2\) term)

Answer:

a.) \(2a^{3}-4a^{2}+5a + 3\)
b.) \(-5x^{2}y^{2}+x^{3}-3y^{3}+7xy\)