Problem 2 - Algebra 1: Concepts and Skills

Question

Is \(9 x^{2}+8 x-4 x^{3}+3\) a polynomial with a degree of \(2 ?\) Explain.

Step-by-Step Solution

Answer

No, the given polynomial \(9 x^{2}+8 x-4 x^{3}+3\) is not of degree 2 but rather of degree 3.

Step 1: Identify the polynomial

Firstly, you need to recognize the given expression, \(9 x^{2}+8 x-4 x^{3}+3 \) as a polynomial.

Step 2: Arrange polynomial in decreasing order

Next, rearrange the polynomial using ascending or descending powers of the variable. For the given polynomial, rearranging in descending order would look like -\(-4 x^{3}+9 x^{2}+8 x+3\). This is because the power of x in the first term is 3, in the second term it's 2, in the third term it's 1 and in the final term there is no x which can be considered as \(x^{0}\). So, while rearranging, we should keep the term with the highest power first.

Step 3: Identify the degree of the polynomial

Finally, look at the rearranged polynomial and identify the degree. The degree of a polynomial is the exponent of the highest degree term. In this case, the first term after rearranging (which is in descending order) is \(-4 x^{3}\) and the exponent of x is 3. So, the degree of the polynomial is 3, not 2.