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Exercise 12.2 class 8 solution- factorisation using algebric identities
To factorize expressions using algebraic identities, you need to be familiar with these identities and apply them appropriately. Here are some common algebraic identities and examples of how to use them for factorization:
These are some common algebraic identities and techniques for factorization. Depending on the expression you’re trying to factor, you may need to use one or more of these identities or apply different techniques. Factorization often requires practice to recognize which identity or method is most suitable for a given expression.
Example: factorise x^2−2xy+y^2−z^2
Exercise 12.2 class 8 solution-factors of the form (x + a)(x + b)
The expression of the form (x + a)(x + b) represents a quadratic equation. The factors of this form can be expanded or factored further. This form is known as a quadratic expression in factored form, and it’s also called a binomial expression. To expand it, you can use the distributive property:
(x + a)(x + b) = x(x) + x(b) + a(x) + a(b) = x^2 + (a + b)x + ab
So, the factors of the form (x + a)(x + b) expand into the quadratic expression x^2 + (a + b)x + ab. In this expanded form, you can see that ‘x^2’ is the leading term, ‘x’ is the linear term, and ‘ab’ is the constant term.
Conversely, if you have a quadratic expression in the form of x^2 + (a + b)x + ab, you can factor it back into the form (x + a)(x + b) by looking for values of ‘a’ and ‘b’ such that when you expand the expression, it matches the given quadratic equation. This process is known as “factoring a quadratic equation.”
For example, if you have the expression x^2 + 5x + 6, you can factor it by finding values of ‘a’ and ‘b’ that satisfy (x + a)(x + b) = x^2 + 5x + 6. In this case, a = 2 and b = 3, so the factorization is (x + 2)(x + 3).
Example 2.Factorize- x2+14x+33,
solve.To factor the quadratic expression x2+14x+33we need to find two numbers that multiply to the constant term (33) and add up to the coefficient of the middle term (14).
The two numbers that satisfy these conditions are 11 and 3 because:
- 11 * 3 = 33
- 11 + 3 = 14
Now, we can use these numbers to factor the expression:
So, the factored form of x2+14x+33 is (x+11)(x+3)
Example 2.To factor the quadratic expression y2−4y−45
solve.you need to find two numbers that multiply to the constant term (-45) and add up to the coefficient of the middle term (-4). These numbers are -9 and 5 because:
- (-9) * 5 = -45
- (-9) + 5 = -4
Now, you can use these numbers to factor the expression:
So, the factored form of y2−4y−45 is (y−9)(y+5)(y−9)(y+5).