Suppose we wish to determine whether or not two given polynomials with complex coefficients have a common root. Given two first-degree polynomials a0 + a1x and b0 + b1x, we seek a single value of x such that
Solving each of these equations for x we get x = -a0/a1 and x = -b0/b1 respectively, so in order for both equations to be satisfied simultaneously we must have a0/a1 = b0/b1, which can also be written as a0b1 - a1b0 = 0. Formally we can regard this system as two linear equations in the two quantities x0 and x1, and write them in matrix form as
Hence a non-trivial solution requires the vanishing of the determinant of the coefficient matrix, which again gives a0b1 - a1b0 = 0.
Now consider two polynomials of degree 2. In this case we seek a single value of x such that
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First one (on the left) is 2) associative property because we are regrouping terms or re-associating them based on the parenthesis. Think of the parenthesis like a tent or a house that separates off the stuff outside. First 27 and 36 are grouped, then we move the parenthesis so that 52 and 27 are grouped now.
Second one (on the right) is 1) commutative. The commutative multiplication property says we can multiply two numbers in any order. For example, 4*9 = 36 and 9*4 = 36. The 4 and 9 swap places. In this problem, the 3 and 4x swap places. The use of parenthesis isn't necessary, though it is sometimes used to imply multiplication.
The answer is 160 because 312÷2 equals 156 and 156 is closer to 160 than 150
