Factors of the form (x + a)(x + b)
This section elaborates on the procedures to factorize algebraic expressions composed of a single variable, particularly those expressed in the format equivalent to the quadratic identity (x + a)(x + b) = x² + (a + b)x + ab. The focus lies on identifying relevant coefficients from the standard expanded form and matching them to the factors of the constant term (ab).
For instance, consider the expression x² + 5x + 6. Here, the product ab equals 6, and the sum a + b must equal 5. Through the exploration of different factors, students can discover that using 2 and 3 satisfies both conditions—yielding the factorized form (x + 2)(x + 3). This section also implies that for expressions with negative or varied coefficients, a careful approach is required to factor them systematically.
Example :
Factorise \( x^2 + 7x + 10 \)
Solution: If we compare the R.H.S. of Identity (IV) with \( x^2 + 7x + 10 \), we find \( ab = 10 \), and \( a + b = 7 \). From this, we must find \( a \) and \( b \). The factors then will be \( (x + a)(x + b) \).
If \( ab = 10 \), it means that \( a \) and \( b \) are factors of 10. Let us take \( a = 5 \), \( b = 2 \). For these values \( a + b = 7 \), and this choice is correct.
Let us try \( a = 10 \), \( b = 1 \). For this, \( a + b = 11 \) which is not exactly required.
The factorised form of this given expression is then \( (x + 5)(x + 2) \).