MarkFL said:You could write:
\(\displaystyle 6x^4+17x^3-24x^2-53x+30=\left(6x^4+12x^3\right)+\left(5x^3+10x^2\right)-\left(34x^2+68x\right)+\left(15x+30\right)=6x^3(x+2)+5x^2(x+2)-34x(x+2)+15(x+2)=(x+2)\left(6x^3+5x^2-34x+15\right)\)
\(\displaystyle 6x^3+5x^2-34x+15=\left(6x^3+18x^2\right)-\left(13x^2+39x\right)+\left(5x+15\right)=6x^2(x+3)-13x(x+3)+5(x+3)=(x+3)\left(6x^2-13x+5\right)\)
\(\displaystyle 6x^2-13x+5=(2x-1)(3x-5)\)
And so we have:
\(\displaystyle 6x^4+17x^3-24x^2-53x+30=(x+2)(x+3)(2x-1)(3x-5)\)
paulmdrdo said:What technique did you use to determine the pair of numbers to be used to rewrite the expression? Was it by trial and error?
A polynomial is an algebraic expression that consists of variables and coefficients, combined using only the operations of addition, subtraction, and multiplication. Examples of polynomials include 2x^3 + 5x^2 - 3x + 1 and 4y^2 - 9.
The degree of a polynomial is the highest exponent of the variable in the expression. For example, the polynomial 2x^3 + 5x^2 - 3x + 1 has a degree of 3, while the polynomial 4y^2 - 9 has a degree of 2.
The process of solving a polynomial equation, also known as finding the roots or zeros of a polynomial, involves determining the values of the variable that make the equation true. This can be done by factoring the polynomial, using the quadratic formula, or using other methods such as the rational root theorem.
Finding the roots of a polynomial is important because it allows us to determine the solutions to the equation. These solutions can be used to solve real-world problems, to graph the polynomial, or to simplify the expression.
The fundamental theorem of algebra states that every polynomial equation with complex coefficients has at least one complex root. This means that for every polynomial of degree n, there are exactly n complex roots. Additionally, the theorem states that every polynomial can be factored into linear and quadratic terms.