Factor Theorem The factor theorem is derived from the remainder theorem. It tells us that there is a relation between factors of the polynomial and zeros of the polynomial. This theorem helps us to find the roots of the polynomial. It also helps us to solve the polynomials of higher degree. Statement: If p(x), a polynomial in x is divided by x-a and the remainder = p (a) = zero, then (x-a) is a factor of p(x). Factor Theorem ProofBack to Top Below you can see the factor theorem proof When p(x) is divided by x - a, Know More About Factoring Worksheets
R = p(a) (by remainder theorem) p(x) = (x-a).q(x)+p(a) (Dividend = Divisor x quotient + Remainder Division Algorithm) But p(a) = 0 is given. Hence p(x) = (x-a).q(x) Conversely if x-a is a factor of p(x) then p(a)=0. p(x) = (x-a).q(x) + R If (x-a) is a factor then the remainder should be zero (x-a divides p(x) exactly) R=0 By remainder theorem, R = p(a) p(a)=0 Learn More About Percent Worksheets
Below you can see the examples on factor theorem Factor Theorem Examples Below you could see the factor theorem examples Example 1: Determine whether x - 2 is a factor of x2-7x+10. Solution p(x) = x2-7x+10 is divided by x - 2. R = p(2) = 4-14+10=0, R=0
Monomials Introduction to monomials In Algebra, a monomial or a term is comprised of a combination of the following: numbers, variables, and exponents. In Algebraic expressions and equations, terms or monomials are separated by addition and subtraction signs Monomial is an algebraic expression with only one term. For example, 7xy, - 5m, 3z2, 4 etc. It contains a constant and variables. We use letters x, y, l, m, ... etc. to denote variables. A variable can take various values. Its value is not fixed. On the other hand, a constant has a fixed value. Examples of constants are: 4, 100, - 17, etc. Examples of MonomialsBack to Top 1. 15xy Coefficient: 15 , variables x and y and exponent 1
2. -2ab2 Coefficient: -2 , variables are a and b and exponent is 2 3. 41pq3 Coefficient: 41 , variables are p and q and exponent is 3 4. -a2 Coefficient: -1 because -a2 is the same as -1a2 Variable is a and the exponent is 2 Addition and Subtraction of MonomialsBack to Top When terms or monomials contain the same variable and same exponent, they are like terms. Addition and subtraction of monomials is done by combining the like terms. Simplify the following expressions. 1) 7 + 7x +13x 2) -12c + 12c 3) 8y - 3y Read More About Trigonometry Worksheets
4) x2 + y2 + x 5) 4np3 - 8np3 Answers : 1) 20x + 7 2) 0 3) 5y 4) x2 + y2 + x 5) -4np3 Multiplication of Monomials When we multiply the monomial, first step is multiplying the numerical coefficients (for e.g. 4 and the 8) and then multiplying the literal coefficients or variables (a and b). Next step is to multiply the like variables by adding their exponents (for e.g. 3+2). (Rule am * an = am+n).
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