The sum and difference of two expressions are 5x 2− x − 4 and x 2+ 9x − 10 respectively. The HCF of the two expressions will be
(x − 1)
Finding the HCF of Two Expressions from their Sum and Difference
The problem provides us with the sum and difference of two algebraic expressions. Let these two expressions be represented by \(A\) and \(B\). We are given:
- Sum of the expressions: \(A + B = 5x^2 - x - 4\)
- Difference of the expressions: \(A - B = x^2 + 9x - 10\)
Our goal is to find the Highest Common Factor (HCF) of expressions \(A\) and \(B\).
Step 1: Find the Expressions A and B
We can find the individual expressions \(A\) and \(B\) by adding and subtracting the given equations:
Adding the two equations:
\((A + B) + (A - B) = (5x^2 - x - 4) + (x^2 + 9x - 10)\)
\(2A = (5x^2 + x^2) + (-x + 9x) + (-4 - 10)\)
\(2A = 6x^2 + 8x - 14\)
Dividing by 2, we get expression \(A\):
\(A = \frac{6x^2 + 8x - 14}{2} = 3x^2 + 4x - 7\)
Subtracting the second equation from the first:
\((A + B) - (A - B) = (5x^2 - x - 4) - (x^2 + 9x - 10)\)
\(A + B - A + B = 5x^2 - x - 4 - x^2 - 9x + 10\)
\(2B = (5x^2 - x^2) + (-x - 9x) + (-4 + 10)\)
\(2B = 4x^2 - 10x + 6\)
Dividing by 2, we get expression \(B\):
\(B = \frac{4x^2 - 10x + 6}{2} = 2x^2 - 5x + 3\)
So, the two expressions are \(A = 3x^2 + 4x - 7\) and \(B = 2x^2 - 5x + 3\).
Step 2: Factorize the Expressions
To find the HCF, we need to factorize both expressions.
Factorizing Expression A: \(A = 3x^2 + 4x - 7\)
We look for two numbers that multiply to \(3 \times (-7) = -21\) and add up to the middle coefficient, 4. These numbers are 7 and -3.
\(A = 3x^2 + 7x - 3x - 7\)
\(A = x(3x + 7) - 1(3x + 7)\)
\(A = (x - 1)(3x + 7)\)
Factorizing Expression B: \(B = 2x^2 - 5x + 3\)
We look for two numbers that multiply to \(2 \times 3 = 6\) and add up to the middle coefficient, -5. These numbers are -2 and -3.
\(B = 2x^2 - 2x - 3x + 3\)
\(B = 2x(x - 1) - 3(x - 1)\)
\(B = (2x - 3)(x - 1)\)
The factorized expressions are:
- \(A = (x - 1)(3x + 7)\)
- \(B = (x - 1)(2x - 3)\)
Step 3: Find the HCF
The Highest Common Factor (HCF) is the product of the factors that are common to both expressions. Comparing the factorized forms of \(A\) and \(B\):
- Factors of A: \((x - 1)\) and \((3x + 7)\)
- Factors of B: \((x - 1)\) and \((2x - 3)\)
The common factor is \((x - 1)\).
Therefore, the HCF of the two expressions is \((x - 1)\).
Conclusion
The HCF of the two expressions is \((x - 1)\).
| Expression | Factorized Form |
|---|---|
| \(A = 3x^2 + 4x - 7\) | \((x - 1)(3x + 7)\) |
| \(B = 2x^2 - 5x + 3\) | \((x - 1)(2x - 3)\) |
The common factor is \((x - 1)\), which is the HCF.
Revision Table: Key Concepts
| Concept | Description | Application in this problem |
|---|---|---|
| Sum and Difference Method | Adding and subtracting two equations involving sum and difference to find individual components. | Used to find the expressions \(A\) and \(B\) from their sum and difference. |
| Algebraic Factorization | Breaking down a polynomial into a product of simpler polynomial factors. | Used to find the factors of \(A\) and \(B\). |
| Highest Common Factor (HCF) | The largest polynomial that divides two or more polynomials without leaving a remainder. It is the product of all common factors. | The final step to identify the common factor \((x - 1)\) from the factorized forms of \(A\) and \(B\). |
Additional Information: Understanding HCF of Polynomials
The HCF of polynomials works similarly to finding the HCF of numbers. For example, the HCF of 12 and 18 is 6. We find this by factoring: \(12 = 2^2 \times 3\) and \(18 = 2 \times 3^2\). The common factors with the lowest powers are \(2^1\) and \(3^1\). So, HCF = \(2 \times 3 = 6\).
For polynomials, we factorize them into irreducible factors over a given field (usually rational numbers for problems like this). The HCF is the product of all common factors, raised to the lowest power they appear in any of the polynomials.
In this problem, the irreducible factors of \(A\) are \((x - 1)\) and \((3x + 7)\). The irreducible factors of \(B\) are \((x - 1)\) and \((2x - 3)\). The only common factor is \((x - 1)\), and it appears with a power of 1 in both cases. Thus, the HCF is \((x - 1)\).
Understanding factorization is crucial for finding the HCF of polynomials. Common techniques include factoring out common monomials, grouping terms, and using specific formulas like the difference of squares or perfect square trinomials, or like in this case, factoring quadratic trinomials by splitting the middle term.
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