Unveiling the Mystery of Factorising: A Walkthrough of x^3 + 3x^2 + 9x+ 5

Factorisation, the process of expressing a polynomial as a product of simpler factors, serves as a cornerstone in algebraic manipulations. It provides a potent tool for solving equations, simplifying expressions, and gaining profound insights into the behavior of polynomials.

Embarking on a Factorisation Adventure

With our sights set firmly on x^3 + 3x^2 + 9x + 5, let us initiate our factorisation adventure by searching for common factors among its terms. Upon inspection, we notice that each term is divisible by x, suggesting that we can factor out an

x as our first step:

x^3 + 3x^2 + 9x + 5 = x(x^2 + 3x + 9 ) + 5

Now, our focus shifts towards factorising the remaining quadratic expression, x^2 + 3x + 9. To achieve this, we employ a technique called completing the square, a powerful strategy for converting any quadratic expression into a square perfect form.

Proceeding with our calculations, we add and subtract the square of half the coefficient of x, which is (3/2)^2, inside the parenthesis:

x^2 + 3x + 9 = x^2 + 3x + (3/2)^2 – (3/2)^2 + 9

Simplifying further, we obtain:

x^2 + 3x + 9 = (x + 3/2)^2 – 9/4 + 9

Combining like terms, we arrive at:

x^2 + 3x + 9 = (x + 3/2)^2 + 5/4

Unveiling the Final Factorisation

Substituting our factorised quadratic expression back into the original equation, we reach the final stage of our factorisation journey:

x^3 + 3x^2 + 9x + 5 = x(x + 3/2)^2 + 5/4 + 5

Merging the constants and simplifying, we arrive at the complete factorisation:

x^3 + 3x^2 + 9x + 5 = (x + 3/2)(x^2 + 3x + 5/4)

Image:

X 3 3x 2 9x 5 Factorise

Conclusion

Through a systematic and logical approach, we have successfully factorised x^3 + 3x^2 + 9x + 5. The factorised form reveals critical insights into the polynomial’s behavior, facilitating further algebraic operations and empowering us to engage with the world of algebra with greater confidence and proficiency. As you continue your mathematical pursuits, may this experience inspire you to embrace the challenges of algebra, unlocking its secrets through the power of exploration and understanding.