Answer to Question #240908 in Calculus for CCW

Question

Factoring Polynomials Practice:

Use synthetic division to divide the polynomials by possible roots

2) 4x³ - 9x² + 6x - 1

P= Q=

Answer 1

Let's consider the polinomial $4x^2-9x^2+6x-1$ . If the zeros of this are rational, then the roots of

polynomial can be sought in the form of fractions, the numerator of which is the divisor of the free term (-1), and the denominator is the divisor of the coefficient at the highest degree (4). Therefore, the roots can be found among the numbers $\pm1; \pm\frac{1}{2}; \pm\frac{1}{4}$ .

If we divide our polynomial $4x^2-9x^2+6x-1$ by the polynomial $(x-1)$ , then we will see that there will be no remainder, but we will get a quotient $4x^2-5x+1$ , which means that $x=1$ is a root of the polynomial. If we divide a second time, we also get the full quotient of the remainder. This means that $x=1$ is the second order root of the polinomial. The remaining quotient $(4x-1)$ takes on a value of zero at $x=\frac{1}{4}.$ The detailed division process can be seen in the photo. So the roots of polinomial are $\{1;\frac{1}{4}\}.$