Factoring Cubic 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 Dividing Fractions 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 – 6x2 + 4x. Solution :- Factors of 2x3 are 1, 2, x, x2, x3,2x, 2x2, 2x3 Learn More About Factoring Trinomials Worksheet
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. 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)
linear equations in two variables Graphing linear equation in two variables is as similar as graphing any linear equation so before learning about linear equations in two variables, the basic knowledge includes about constructing linear equations graph and what is rectangular axes etc. Rectangular Axes for Linear Equation Graph The position of a point in a plane is fixed by selecting two axes of reference which are formed by combining two number lines at right angles so that their zeros coincide. The horizontal number line is called x-axis and the vertical number line is called y-axis. The point of intersection of the two number lines is called origin. The two number lines together are called rectangular axes.
Co-ordinates in Rectangular Axes The position of a point with respect to the rectangular axes by means of a pair of numbers is called co-ordinates. The distance OM of point P along x-axis is called x-co-ordinate or abscissa. The distance ON of point P along y-axis is called ordinate or y-co-ordinate. If OM=a and ON=b then position of the point P is denoted by (a, b). Note on Co-ordinates: Co-ordinates of the origin is (0, 0). Co-ordinates of any point on the x-axis is (x, 0). Co-ordinates of any point on the y-axis is (0, y). Quadrants of Rectangular Axes The rectangular axes divide the plane into four regions called quadrant. By convention the quadrants are numbered as I, II, III, IV in the Read More About Ordering Fractions Worksheet
Any point in the I quadrant will have both the co-ordinates positive. In the II quadrant, x-co-ordinates is negative while y-co-ordinate positive. In the III quadrant, x-co-ordinate as well as y-co-ordinate both are negative. In the IV quadrant, x-co-ordinate is positive while the y-co-ordinate is negative. Plotting Graph and Calculating Area for Linear Equation Values Plot the graph of x+3y=6. Use the graph to find a) area between the line and axes b) value of y when x=-6 Answer: x+3y=6
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