Factor Polynomials Factoring Polynomials refers to factoring a polynomial into irreducible polynomials over a given field. It gives out the factors that together form a polynomial function. A polynomial function is of the form xn + xn -1 + xn - 2 + . . . . + k = 0, where k is a constant and n is a power. Polynomials are expressions that are formed by adding or subtracting several variables called monomials. Monomials are variables that are formed with a constant and a variable of some degree. Examples of monomials are 5x3, 6a2. Monomials having different exponents such as 5x3 and 3x4 cannot be added or subtracted but can be multiplied or divided by them. Any polynomial of the form F(a) can also be written as F(a) = Q(a) x D (a) + R (a) using Dividend = Quotient x Divisor + Remainder. Know More About Percent Word Problems
If the polynomial F(a) is divisible by Q(a), then the remainder is zero. Thus, F(a) = Q(a) x D(a). That is, the polynomial F(a) is a product of two other polynomials Q(a) and D(a). For example, 2t + 6t2 = 2t x (1 + 3t). Variables, Exponents, Parenthesis and Operations (+, -, x, /) play an important role in factoring a polynomial. Factorization by dividing the expression by the GCD of the terms of the given expression: GCD of a polynomial is the largest monomial, which is a factor of each term of the polynomial. It involves finding the Greatest Common Divisor (GCD) or Highest Common Factor (HCF) of the terms of the expression and then dividing each term by its GCD. Therefore the factors of the given expression are the GCD and the quotient thus obtained. Example 1: Factorize : 2x3 â€“ 6x2 + 4x. Solution: Factors of 2x3 are 1, 2, x, x2, x3,2x, 2x2, 2x3 Factors of 6x2 are 1, 2,3, 6, x, x2, 2x, 2x2 ,3x, 3x2 ,6x, 6x2 Learn More About Adding and Subtracting Fractions Worksheet
Factors of 4x are 1,2,4,x,2x,4x. Thus the GCD of the above terms is 2x. Dividing 2x3 , -6x2 and 4x by 2x, we get x2 - 3x + 2 Then the GCD becomes one factor and the quotient is the other factor. 2x3 – 6x2 + 4x = 2x .(x2 - 3x + 2) Therefore the factors of 2x3 – 6x2 + 4x are 2x and (x2 - 3x + 2) Thus, 2x3 – 6x2 + 4x = 2x .(x2 - 3x + 2) Factorization by grouping the terms of the expression: Grouping the terms of the expression in such a way that there are common factors among the terms of the groups so formed.
Linear Combination Linear combination is combination of two lines. Linear combination is also called as addition method. Linear combination is one of the methods to solve system of two equations. By this method, we can solve the variable x and y. Steps for solving systems of equation using linear combination method: We need to add two line equations for eliminating any one variable. Then we get a new equation of one variable. Solve this, we get a value of that variable. Substitute this variable value into any one original equation, we get a value for another variable. Let us see about how to solve linear equations using linear combination method. Solving Linear Combination Below you could see examples for solving linear combination
Example 1: Solving the linear equations using linear combination method. 7X - y = 5 and 2x + 3y = 8 Solution :- The given equations are 7X - y = 5 (1) 2x + 3y = 8 (2) Multiply by 3 to the equation (1) 21x - 3y = 15 Multiply by 1 to the equation (2) 2x + 3y = 8 Add both these equations. 21x - 3y + (2x + 3y) = 15 + 8 21x - 3y + 2x + 3y = 15 + 8 Read More About Algebra Word Problems Worksheet
Combine like terms. 21x + 2x - 3y + 3y = 23 23x = 23 Isolate the variable x. X = 1 Substitute the value of x variable into the equation (2) 2X + 3y = 8 2(1) + 3y = 8 2 + 3y = 8 Subtract 2 from each side. 2 - 2 + 3y = 8 - 2 3y = 6 Isolate the variable y. Y=2 Therefore, the solutions are x = 1 and y = 2.