Values Of A Polynomial FunctionFor a polynomial f(x) = 3x2 – 4x + 2. Values Of A Polynomial To find its value at x = 3; replace x by 3 everywhere. So, the value of f(x) = 3x2 – 4x + 2 at x = 3 is f(3) = 3 × 32 – 4 × 3 + 2 = 27 – 12 + 2 = 17. Similarly, …

Factor TheoremTheorem: If p(x) is a polynomial of degree n ≥ 1 and a is any real number, then (i) x – a is a factor of p(x), if p(a) = 0, and (ii) p(a) = 0, if x – a is a factor of p(x). Proof: By the Remainder Theorem, p(x) = (x – a) q(x) + p(a). (i) I…

Types Of Factorization Example Problems With SolutionsType I: Factorization by taking out the common factors.  Example 1:    Factorize the following expression 2x2y + 6xy2 + 10x2y2 Solution:    2x2y + 6xy2 + 10x2y2 =2xy(x + 3y + 5xy) Type II: Factorizati…

Factorization Of Algebraic Expressions Of The Form a3 + b3 + c3, When a + b + c = 0Example 1:   Factorize (x – y)3 + (y – z)3 + (z – x)3 Solution:    Let x – y = a, y– z = b and z – x = c, then, a + b + c = x – y + y – z + z –x = 0 ∴ a3 + b3 + c3 = 3abc …

Relationship Between Zeros And Coefficients Of A PolynomialConsider quadratic polynomial P(x) = 2x2 – 16x + 30. Now, 2x2 – 16x + 30 = (2x – 6) (x – 3) = 2 (x – 3) (x – 5) The zeros of P(x) are 3 and 5. Sum of the zeros = 3 + 5 = 8 = \(\frac { -\left( -16…

Factorization Of Polynomials Using Factor TheoremFactor Theorem:If p(x) is a polynomial of degree n  1 and a is any real number, then (i) x – a is a factor of p(x), if p(a) = 0, and (ii) p(a) = 0, if x – a is a factor of p(x). Proof: By the Remainder Th…

Form A Polynomial With The Given ZerosLet zeros of a quadratic polynomial be α and β. x = β,               x = β x – α = 0,   x ­– β = 0 The obviously the quadratic polynomial is (x – α) (x – β) i.e.,  x2 – (α + β) x + αβ x2 – (Sum of the zeros)x + Prod…

Degree Of A PolynomialThe greatest power (exponent) of the terms of a polynomial is called degree of the polynomial. For example : In polynomial 5x2 – 8x7 + 3x: (i) The power of term 5x2 = 2 (ii) The power of term –8x7 = 7 (iii) The power of 3x = 1 Since…

Monomials, Binomials, and PolynomialsA monomial is the product of non-negative integer powers of variables. Consequently, a monomial has NO variable in its denominator. It has one term. (mono implies one) (notice: no negative exponents, no fractional ex…

Dividing PolynomialsWe will be examining polynomials divided by monomials and by binomials.Dividing a Polynomial by a Monomial:Steps for Dividing a Polynomial by a Monomial: Divide each term of the polynomial by the monomial. a) Divide numbers (coeffici…