Mada za sehemu hiiAlgebraMada 4
Factorization is the process of resolving an expression into its factors. It involves reversing the operation of expansion, which is done by removing brackets. The reverse operation is called factorizing, and it is done by adding brackets around the factors.
Factorizing linear expressions involves finding common factors in each term and factoring them out.
Example 1: Factorize the expression
Solution: In the expression , we can observe that both terms have a common factor of 5. So, we factor out 5 from both terms:
Example 2: Factorize
Solution: Here, we need to find the highest common factor (HCF) of the two terms. The common factors are 6, , and . So, factoring out gives:
When a quadratic expression is written as the product of two factors, we say that the expression is factorized. There are different methods to factorize quadratic expressions:
- Factorization by splitting the middle term
- Factorization by inspection
This method involves splitting the middle term (the coefficient of in ) into two terms that can be factored.
Example 1: Factorize by splitting the middle term
Solution: We start with the quadratic expression . We need to find two numbers that multiply to and add up to -2. The numbers are 4 and -6, since and . Now, factor by grouping: Factor out the common binomial factor:
Example 2: Factorize by splitting the middle term
Solution: The product of , and the two numbers that multiply to 25 and add up to 10 are 5 and 5. So, we split the middle term as follows: Now, factor by grouping: Factor out the common binomial factor:
In this method, we look at the quadratic expression and directly identify the factors based on the coefficients.
Example 1: Factorize by inspection
Solution: We need two numbers that multiply to and add up to 3. The numbers are 1 and 2, so we can directly write the factors as:
Example 2: Factorize by inspection
Solution: We need two numbers that multiply to and add up to 5. The numbers are 8 and -3. We split the middle term: Now, factor by grouping: Factor out the common binomial factor:
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