Question

students were asked to write $2x^3 + 3x + 4x^2 + 1$ in standard form. four student responses are shown below.\
alexa: $4x^2 + 3x + 2x^3 + 1$\
carol: $2x^3 + 3x + 4x^2 + 1$\
ryan: $2x^3 + 4x^2 + 3x + 1$\
eric: $1 + 2x^3 + 3x + 4x^2$\
which students response is correct?\

1. alexa\
2. carol\
3. ryan\
4. eric

Explanation:

Step1: Recall standard form of polynomial

A polynomial in standard form is written with terms in descending order of their exponents. For a polynomial in one variable \(x\), the standard form is \(a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0\), where \(n\) is the highest degree (exponent) of the variable.

Step2: Determine the degree of each term in the given polynomial

The given polynomial is \(2x^3+3x + 4x^2+1\).

• The term \(2x^3\) has degree \(3\) (exponent of \(x\) is \(3\)).
• The term \(3x\) has degree \(1\) (exponent of \(x\) is \(1\)).
• The term \(4x^2\) has degree \(2\) (exponent of \(x\) is \(2\)).
• The term \(1\) (which can be written as \(1x^0\)) has degree \(0\).

Step3: Arrange terms in descending order of degrees

We need to arrange the terms from the highest degree to the lowest degree.

• Highest degree term: \(2x^3\) (degree \(3\)).
• Next highest degree term: \(4x^2\) (degree \(2\)).
• Next: \(3x\) (degree \(1\)).
• Last: \(1\) (degree \(0\)).

So, the polynomial in standard form should be \(2x^3 + 4x^2+3x + 1\), which is Ryan's response.

Answer:

1. Ryan