Now, we factorise each pair separately:

(x^3-13x^2)=x^2(x-13)

(32x-20)=4(8x-5)

Combining these factors together, we get:

x^3-13x^2+32x-20=(x^2-13)(4x-5)

Method 2: Trial and Error

While grouping led us to the desired result, there’s an alternative approach that involves trial and error.

We look for two binomials that, when multiplied together, produce the original polynomial.

(x – p)(x^2 + px + q)

This approach requires plugging in values for p and q until we find a combination that gives us our target polynomial. After some attempts, we find that:

p=4, q=-5

Substituting these values back into our binomial, we obtain the desired factorised form:

x^3-13x^2+32x-20=(x-4)(x^2+4x-5)

Method 3: Synthetic Division

For more complex polynomials, synthetic division offers a convenient way to find integer roots. We use this approach if we have a factor of the form x-k.

For our polynomial and a potential root of x-4, we set up the synthetic division as follows:

4 | 1 -13 32 -20

\ \ \ 4 -16 64

——————–

\ \ 1 -9 94 44

The last entry in the bottom row (44) gives us a remainder of 0, confirming that x-4 is indeed a factor of our polynomial.

Using this factor, we repeat the process to find the remaining factor:

-5 | 1 -9 94 44

\ \ \ -5 50 -220

——————–

\ \ 1 -14 44 -176

The last entry (0) confirms that x-(-5)=x+5 is the other factor, resulting in the same factorised form:

x^3-13x^2+32x-20=(x-4)(x+5)

Tips and Expert Advice

Mastering the art of factorisation requires practice and understanding. Here are some expert tips to enhance your skills:

• Start with simpler polynomials: Before tackling complex problems, gain comfort with factorising simple polynomials to build a solid foundation.
• Look for patterns: Identify patterns within the coefficients to make your factorisation process more efficient.
• Break down large polynomials: If you’re faced with a large polynomial, break it down into smaller chunks to make it more manageable.
• Use a variety of methods: Don’t limit yourself to one method. Explore different approaches to find the one that suits you best.
• Check your work: Always multiply your factors back together to ensure you’ve factorised the polynomial accurately.

By following these expert recommendations, you’ll become more proficient in factorising polynomials, broadening your mathematical horizons.

Frequently Asked Questions

As we conclude our factorisation adventure, let’s address some common questions that may linger in your mind:

• Q. What is the purpose of factorisation?

• A. Factorisation simplifies polynomials, making them easier to solve and understand. It also aids in finding roots, solving simultaneous equations, and more.

• Q. Are there different methods for factorisation?

• A. Yes, various factorisation methods exist, including grouping, trial and error, synthetic division, and using the zero product property.

• Q. How do I know if a polynomial is factorised completely?

• A. Multiply the factors back together. If you get the original polynomial, then the factorisation is complete.

Factorise X 3 13x 2 32x 20

Conclusion

Through this comprehensive journey, we’ve delved into the realm of polynomial factorisation, exploring various methods to tackle the problem of x^3-13x^2+32x-20. Remember, practice makes perfect, so keep honing your factorisation skills to conquer even more complex equations.

Are you eager to dive deeper into the world of polynomials and factorisation? Share your thoughts and questions in the comments below, and let’s continue this mathematical exploration together!