Write in Standard Form x/4-2x^5+(x^3)/2+1
The provided problem is asking for the expression x/4 - 2x^5 + (x^3)/2 + 1 to be rewritten in standard form. Standard form for a polynomial means arranging the terms in descending order of their exponents, and combining any like terms if necessary. Each term consists of a coefficient (which might be a fraction) and a variable raised to a power (the exponent). To put the expression in standard form, one would typically start with the term with the largest exponent and proceed in order to the term with the smallest exponent or constant term.
$\frac{x}{4} - 2 x^{5} + \frac{x^{3}}{2} + 1$
Answer
Solution:
Step 1:
To express a polynomial in the standard form, we first combine like terms and then arrange the terms from the highest degree to the lowest degree, in the form $a x^{n} + b x^{n-1} + ... + d$.
Step 2:
Shift the term $\frac{x}{4}$ to its correct position in the sequence. The polynomial becomes $- 2 x^{5} + \frac{x^{3}}{2} + \frac{x}{4} + 1$.
Step 3:
Rearrange the polynomial in descending powers of $x$. The standard form is $- 2 x^{5} + \frac{1}{2} x^{3} + \frac{1}{4} x + 1$.
Knowledge Notes:
To write a polynomial in standard form, you need to follow these guidelines:
Combining Like Terms: If there are any like terms in the polynomial (terms with the same variable raised to the same power), they should be combined by adding or subtracting the coefficients.
Ordering Terms: The terms should be arranged in descending order of their degree. The degree of a term is the exponent of the variable. For example, in the term $a x^{n}$, $n$ is the degree.
Standard Form: The standard form of a polynomial with one variable $x$ is written as $a_n x^{n} + a_{n-1} x^{n-1} + ... + a_1 x + a_0$, where $a_n, a_{n-1}, ..., a_1, a_0$ are coefficients, and $n$ is a non-negative integer. The term with the highest degree is written first and is followed by terms with lower degrees in descending order.
Coefficients: Coefficients are the numerical factors in front of terms containing variables. In the standard form, coefficients are often written as fractions or whole numbers.
Constants: The constant term (without any variable) is written at the end of the polynomial.
Fractional Coefficients: If there are fractional coefficients, they should be simplified if possible, but it's acceptable to leave them as fractions in the final expression.
Variable Terms: When arranging the terms, only the exponents of the variable are considered for the order, regardless of the coefficients.
By following these steps, you can write any polynomial in its standard form, which is useful for various algebraic operations and for understanding the structure of the polynomial.
