How do you 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 Comparing Decimals Worksheet
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 Elapsed Time 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)
Graph Linear Equations What is a linear equation: An equation is a condition on a variable. A variable takes on different values; its value is not fixed. Variables are denoted usually by letter of alphabets, such as x, y , z , l , m , n , p etc. From variables we form expression. Linear equation in one variable: These are the type of equation which have unique (i.e, only one and one ) solution. For example: 2 x + 5 = 0 is a linear equation in one variable. Root of the equation is âˆ’52 Example 1: Convert the following equation in statement form. x-5=9 Solution : 5 taken from x gives 9
So x = 9 + 5 = 14 Hence, x = 14 For verification of the statement, x-5=9 14 - 5 = 9 9 = 9 So left hand side value is equal to right hand side value. Hence the value of x determined is correct . You can try out some more examples from linear equations worksheets Linear equation in two variable: An equation which can be put in the form ax+by+c=0, where a, b, and c are real numbers, and a and b are not zero, is called linear equation in two variables. For example: 3 x + 4 y = 8 which is a equation in two variables. Summary:
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A linear equation in two variable has infinitely many solutions.The graph of every linear equation in two variable is a straight line. Every point on the graph of a linear equation in two variable is a solution of the linear equation. An equation of the type y = mx represents a line passing through the origin. How to Solve Linear EquationsBack to Top Below are the methods for solving linear equations in one variable: Method 1: Isolate the variable: In this method we will isolate the variable on one side and number on other sides. Steps and example for solving equation: Example 1: solve 2x + 3 =15 Solution 1: Given equation is: 2x + 3 = 15 Step 2: Subtract 3 from both side 2x + 3 - 3 = 15 - 3 2 x =12
What is a linear equation: An equation is a condition on a variable. A variable takes on different values; its value is not fixed. Variables...