Factoring 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 Multiplying Fractions Worksheets
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 – 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 Factors of 4x are 1,2,4,x,2x,4x. Learn More About Factoring Polynomials Worksheet
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. Example: Factorize 3x + xy + 3y + y2 Hint: Notice that there is no factor common to all the terms. So regroup the terms of the expression. In this expression, there is a common factor for the first two terms. Similarly the last two terms have a common factor. Therefore 3x + xy + 3y + y2= x (3 + y) + y ( 3+y) 3x + xy + 3y + y2 = ( 3 + y) ( x + y) This can also be regrouped in another way,
Standard Form Standard form is generally a syntax kind for expressing mathematical operations. Learn about the concept here or you can also connect to an online tutor anytime and thus gain your answers to math problems regarding standard form. Get your help now. Below is explained about standard form in math, algebra and equations. What is Standard Form? The definition of Standard form is that it is a way of writing down very large or very small numbers easily. 103 = 1000, so 4 x 103 = 4000. So 4000 can be written as 4 × 103. Standard form is also used to write even larger numbers down easily in standard form. Small numbers can also be written in standard form. Example: Write 50 400 000 000 000 in standard form: 50 400 000 000 000 = 5.04 × 1013 It’s 1013 because the decimal point has been moved 13 places to the left to get the number to be 5.04
Standard Form in Algebra Standard form algebra is used to write down the complex equations in a simple form i.e. to write a large equation very easily. For example standard form of a linear equations is ax + by + c= 0, standard form of quadratic equation is ax2 + bx + c = 0. Standard form algebra is used to find out the factors. Example: 3x2 + 11x - 4 = 0 [3 x 4 = 12] 12 - 1 = 11(since 11 is b) 3x2 + 12x - x - 4 = 0 3x (x + 4) -1 (x + 4) So the factors are (3x - 1) and (x + 4). xy + 3y - 2x - 6 = 0 arrange them in an order now. We get xy - 2x + 3y - 6 = 0 Read More About Least Common Multiple Worksheet
x (y - 2) + 3 (y - 2) = 0 (x + 3) (y - 2) = 0 So here (x + 3) and ( y - 2) are the factors. Standard Form Equation Standard form equation is the general representation of an equation. There are different types of standard form equation such that linear equation, quadratic equation, polynomials etc. Example: Find the equation of a line if slope is 2 and passing through a point (2, 4) Given that Slope m = 2 Point = (2, 4) Equation of a line passing through a point and with slope m is y - y0 = m (x - x0) y - y0 = m (x - x0) y - 4 = 2 (x – 2),
y - 4 = 2x - 4
2x - 4 - y + 4 =0,
2x - y - 4 + 4 =0,
Therefore equation of a line is 2x - y = 0.
2x - y =0
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