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## Program Code โ Mastering Polynomial Factoring: A Comprehensive Guide

```
``````
import sympy as sp
def factor_polynomials_3_terms(expression):
'''
This function factors a polynomial with 3 terms using sympy.
Parameters:
expression (str): A string expression of the polynomial to be factored.
Returns:
factored_expression (str): The factored form of the input polynomial, as a string.
'''
x = sp.symbols('x') # Define the symbol
polynomial = sp.sympify(expression) # Convert the string expression into a sympy expression
# Factor the polynomial
factored_expression = sp.factor(polynomial)
# Convert the factored expression back into a string for display
return str(factored_expression)
# Example polynomial with 3 terms
expression = 'x**2 - 5*x + 6'
factored_expression = factor_polynomials_3_terms(expression)
print('Factored Expression:', factored_expression)
```

### ### Code Output:

Factored Expression: (x โ 3)*(x โ 2)

### ### Code Explanation:

This program is designed to factor polynomials, specifically those with three terms, leveraging the power of the `sympy`

library in Python. The concept is straightforward but incredibly efficient, especially for those who grapple with algebra every so often.

Firstly, we import the `sympy`

library, which is essentially our best friend in this journey. Itโs a Python library for symbolic mathematics, offering a vast array of functionalities for simplification, expansion, equations solving, and yes, factorization!

We define a function `factor_polynomials_3_terms`

that takes one argument: `expression`

. This `expression`

is a string representing the polynomial youโre itching to factor. Inside, we initiate a symbolic variable `x`

because, well, most polynomials are fond of `x`

.

Next, we use `sympy.sympify`

to transform the string expression into something `sympy`

can work its magic on โ a symbolic expression. Why? Because strings are for humans, and sympy expressions are for computational wizards.

Then, the showโs star: `sp.factor(polynomial)`

. Itโs like asking โPretty please, can you simplify this mess for me?โ and `sympy`

graciously obliges, handing back the polynomial in its factored form. Neat, huh?

Finally, we convert this factored expression back into a string, because we humans love reading those, and return it.

In the provided example, weโre dealing with `x**2 - 5*x + 6`

. Itโs called into the function, and voilร , the output is `(x - 3)*(x - 2)`

, showcasing that the original polynomial can be expressed as a product of two polynomials.

This program not only makes polynomial factorization a breeze but also exemplifies the marvellous synergy between programming and mathematics. Who said math had to be a dull affair?

### Frequently Asked Questions

**What are three-term polynomials?**- Three-term polynomials are algebraic expressions consisting of three different terms, typically in the form (ax^2 + bx + c), where (a), (b), and (c) are constants and (x) is a variable.

**Why is factoring three-term polynomials important in algebra?**- Factoring three-term polynomials is crucial in algebra as it allows for simplification of expressions, solving equations, and identifying patterns that can help in further algebraic manipulations.

**What are the common methods for factoring three-term polynomials?**- The common methods for factoring three-term polynomials include the distributive property, factoring by grouping, factoring trinomials, factoring by substitution, and recognizing special cases like perfect square trinomials.

**How do you factor the difference of squares in polynomials with three terms?**- To factor the difference of squares in polynomials with three terms, you identify the terms that can be written as the square of a binomial, and then apply the formula ((a^2 โ b^2) = (a + b)(a โ b)) to factorize the expression.

**What are perfect square trinomials, and how do you identify and factor them?**- Perfect square trinomials are trinomials that can be factored into the square of a binomial, such as (a^2 + 2ab + b^2 = (a + b)^2). To identify and factor perfect square trinomials, look for patterns that resemble ((a + b)^2) or ((a โ b)^2) forms.