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What is the HCF of Polynomials by Factorisation?
Grade Level:
Class 7
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The HCF (Highest Common Factor) of polynomials by factorisation is the largest common factor that divides two or more polynomials exactly. It is found by breaking down each polynomial into its simplest multiplicative parts (factors) and then identifying the factors that are common to all of them, taking the lowest power of each common factor.
Simple Example
Quick Example
Imagine you have two friends, Rohan and Priya, who both love collecting cricket cards. Rohan has cards of players whose jersey numbers are multiples of 6 (like 6, 12, 18), and Priya has cards of players whose jersey numbers are multiples of 9 (like 9, 18, 27). The common jersey number they both could have is 18. Similarly, for polynomials, we find the 'common jersey number' by finding their common factors.
Worked Example
Step-by-Step
Let's find the HCF of the polynomials P(x) = 6x^2y and Q(x) = 9xy^2.
Step 1: Factorise the first polynomial, P(x) = 6x^2y.
6 = 2 * 3
x^2 = x * x
y = y
So, P(x) = 2 * 3 * x * x * y
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Step 2: Factorise the second polynomial, Q(x) = 9xy^2.
9 = 3 * 3
x = x
y^2 = y * y
So, Q(x) = 3 * 3 * x * y * y
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Step 3: Identify the common factors in both factorisations.
Common numerical factor: 3 (present in both 6 and 9)
Common variable 'x' factor: x (x is present once in Q(x) and twice in P(x), so we take the lowest power, which is x^1)
Common variable 'y' factor: y (y is present once in P(x) and twice in Q(x), so we take the lowest power, which is y^1)
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Step 4: Multiply the common factors together.
HCF = 3 * x * y
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Answer: The HCF of 6x^2y and 9xy^2 is 3xy.
Why It Matters
Understanding HCF of polynomials is super useful in many advanced fields. In Computer Science, it helps simplify complex code and algorithms, making software run faster. Engineers use it to design efficient systems, from building bridges to creating circuits, by finding common parts. Even in Data Science, it can help in pattern recognition and simplifying data models.
Common Mistakes
MISTAKE: Students often forget to take the lowest power of common variables. For example, for x^3 and x^2, they might take x^3. | CORRECTION: Always take the common factor with the smallest exponent. For x^3 and x^2, the common factor is x^2.
MISTAKE: Not fully factorising the numerical coefficients. For example, for 12 and 18, they might only see 6 as common and not break it down to 2*3. | CORRECTION: Always break down the numbers into their prime factors (e.g., 12 = 2*2*3, 18 = 2*3*3) to ensure you find ALL common numerical factors.
MISTAKE: Including factors that are not common to ALL polynomials. If finding HCF of three polynomials, a factor must be present in all three. | CORRECTION: Double-check that each factor you include in the HCF is indeed a common factor across every single polynomial given.
Practice Questions
Try It Yourself
QUESTION: Find the HCF of 10a^3b and 15a^2b^2. | ANSWER: 5a^2b
QUESTION: What is the HCF of 8x^2y^3z and 12xy^2z^2? | ANSWER: 4xy^2z
QUESTION: Find the HCF of 14p^4q^2r, 21p^3q^3r, and 7p^2q^2r^2. | ANSWER: 7p^2q^2r
MCQ
Quick Quiz
What is the HCF of 4m^2n and 6mn^3?
2mn
12m^2n^3
2m^2n^3
4mn^3
The Correct Answer Is:
A
To find the HCF, we factorise: 4m^2n = 2*2*m*m*n and 6mn^3 = 2*3*m*n*n*n. The common factors are 2, m (lowest power is m^1), and n (lowest power is n^1). So, HCF = 2*m*n = 2mn.
Real World Connection
In the Real World
Imagine you are a software developer creating an app that organises files on a computer. If you have many files with similar naming patterns (like 'ProjectX_report_v1.pdf', 'ProjectX_data_v2.xlsx'), finding the HCF of their 'name polynomials' can help identify the common 'ProjectX' part. This helps in grouping them efficiently, just like how file explorers group similar files together.
Key Vocabulary
Key Terms
POLYNOMIAL: An expression with one or more algebraic terms, each consisting of a constant multiplied by one or more variables raised to non-negative integer powers. | FACTORISATION: The process of breaking down a number or expression into its multiplicative components (factors). | HCF (HIGHEST COMMON FACTOR): The largest factor that two or more numbers or polynomials share. | EXPONENT: The power to which a number or variable is raised, indicating how many times it is multiplied by itself.
What's Next
What to Learn Next
Great job learning HCF of polynomials! Next, you should explore 'LCM (Lowest Common Multiple) of Polynomials by Factorisation'. Understanding LCM will help you further simplify algebraic fractions and solve more complex problems, building on the factorisation skills you just mastered.
