Write in Standard Form 3x+2x^2-4x+3x^2-5x
This problem asks for the simplification of a polynomial expression by combining like terms and reordering it into standard form. Standard form for a polynomial is when the terms are arranged in descending order of their exponents, with the highest degree first. The like terms, which are those with the same variable raised to the same power, need to be added or subtracted from each other accordingly. In this given polynomial, the terms with x^2 need to be combined, as well as the terms with just x. Once combined, they should be placed in the correct order to reflect standard polynomial form.
$3 x + 2 x^{2} - 4 x + 3 x^{2} - 5 x$
Answer
Solution:
Step 1:
First, combine like terms and organize the polynomial in descending powers of $x$. The standard form is $ax^2 + bx + c$.
Step 2:
Combine $3x$ and $-4x$. The expression now looks like $2x^2 - x + 3x^2 - 5x$.
Step 3:
Next, add $2x^2$ to $3x^2$. This simplifies to $5x^2 - x - 5x$.
Step 4:
Finally, combine $-x$ and $-5x$. The polynomial in standard form is $5x^2 - 6x$.
Knowledge Notes:
To write a polynomial in standard form, one must follow certain steps:
Combine like terms: Like terms are terms that have the same variables raised to the same power. In this case, terms with $x^2$ are combined, and terms with $x$ are combined.
Arrange in descending order: The terms should be written starting from the highest power to the lowest power of the variable. The standard form for a quadratic polynomial is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
Simplify the expression: This involves performing addition or subtraction on the coefficients of like terms.
In the given problem, the terms $2x^2$ and $3x^2$ are like terms, as are $3x$, $-4x$, and $-5x$. These are combined to simplify the polynomial to its standard form.
The process of simplifying polynomials is foundational in algebra and is used in various applications, including solving quadratic equations, graphing parabolas, and optimizing functions.
