Factorisation of Polynomials
In this section, we delve into the process of factorising polynomials, a crucial aspect of algebra that simplifies understanding polynomial behavior. The Factor Theorem plays a central role, stating that if a polynomial p(x) of degree n has a real number a such that p(a) = 0, then (x - a) is a factor of p(x). Conversely, if (x - a) is a factor of p(x), then p(a) = 0. This theorem arises from the Remainder Theorem, which establishes that dividing a polynomial by (x - a) yields a remainder of zero when a is a root.
The section also demonstrates factorisation techniques for quadratic polynomials, suggesting methods such as splitting the middle term or using the Factor Theorem. For instance, to factor a polynomial like axΒ² + bx + c, we must express b as the sum of two numbers whose product equals ac.
Examples illustrate these processes, such as checking if x + 2 is a factor of polynomials or identifying roots. By employing these methods, students gain a stronger grasp of polynomial equations and their factorizable forms, paving the way for more advanced algebraic manipulation.
Example: Find the value of \( k \), if \( x - 2 \) is a factor of \( 5x^3 + 4x^2 - 3x - 2 + k \).
Solution: As \( x - 2 \) is a factor of \( p(x) = 5x^3 + 4x^2 - 3x - 2 + k \), \( p(2) = 5(2)^3 + 4(2)^2 - 3(2) - 2 + k = 0 \)
Now,
\[ 5(2)^3 + 4(2)^2 - 3(2) - 2 + k = 0 \]
So,
\[ 5 \cdot 8 + 4 \cdot 4 - 6 - 2 + k = 0 \]
i.e.,
\[ k = -30 \]