Factorization Factor Theorem
The Factor Theorem is an extension of the remainder theorem where the remainder is \(0\).
- \((x-a)\) is a factor of \(p(x)\), if \(p(a) = 0\).
- \(p(a) = 0\), if \((x-a)\) is a factor of \(p(x)\).
Factorization of a Quadratic Polynomial (\(ax^2 + bx + c\)):
Quadratic factorization is performed by splitting the middle term (\(bx\)).
Step - 1: Find the product of the first and last term (\(a \times c\)).
Step - 2: Find two factors of this product whose sum (or subtraction) equals the middle term coefficient (\(b\)).
Step - 3: Rewrite the middle term (\(bx\)) as the sum of the two new factors (e.g., \(6x^2 + 15x + 4x + 10\)).
Step - 4: Group the terms into two pairs and factor out common factors from each pair.
Step - 5: Factor out the common binomial factor to get the final factors.
Factorization of a Cubic Polynomial (\(p(x)\)):
Steps to factorize the cubic polynomial \(p(x)\).
Step - 1: Find a value \(x=a\) such that \(p(a)=0\) (i.e., find one zero/root).
Step - 2: By the Factor Theorem, \((x-a)\) is a factor of \(p(x)\).
Step - 3: Divide \(p(x)\) by \((x-a)\) to get a quotient, which will be a quadratic equation.
Step - 4: Factorize the quadratic quotient by splitting its middle term.
Algebraic Identities:
Identities are algebraic equations that are always true regardless of the values assigned to the variables. They provide an alternative method for solving problems involving multiplication and factorization of algebraic expressions and numbers.
Squaring Identities:
Identity I: \((a+b)^2 = a^2 + 2ab + b^2\)
Identity II: \((a-b)^2 = a^2 - 2ab + b^2\)
Identity III: \((a+b)(a-b) = a^2 - b^2\)
Identity IV: \((x+a)(x+b) = x^2 + (a+b)x + ab\)
Trinomial Expansion:
Identity IV: \((a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\)
Cubing Identities:
Identity VI: \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\) or \(a^3 + b^3 + 3ab(a+b)\)
Identity VII: \((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\) or \(a^3 - b^3 - 3ab(a-b)\)
Factorization Identities (Derived)Sum of Cubes:
Identity VIII: \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)
From \((a+b)^3 = a^3 + b^3 + 3ab(a+b)\), rearrange to \(a^3 + b^3 = (a+b)^3 - 3ab(a+b)\), then factor out \((a+b)\).
Difference of Cubes:
Identity IX: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)
From \((a-b)^3 = a^3 - b^3 - 3ab(a-b)\), rearrange to \(a^3 - b^3 = (a-b)^3 + 3ab(a-b)\), then factor out \((a-b)\).
Three Different Values:
Identity X: \((x+a)(x+b)(x+c) = x^3 + (a+b+c)x^2 + (ab+bc+ca)x + abc\).