3x^3-3x^2-x+3
Factoring
The polynomial 3x^3-3x^2-x+3 can be factored as follows:
(x-3)(3x^2+x+1) To factor this polynomial, we can use the sum-product pattern. The sum of the roots of a polynomial is equal to the negative of the constant term, and the product of the roots is equal to the constant term divided by the leading coefficient. In this case, the constant term is 3, and the leading coefficient is 3. Therefore, the sum of the roots is -3, and the product of the roots is 1.
We can use this information to write down two equations:
r_1 + r_2 + r_3 = -3 r_1 r_2 r_3 = 1 We can solve these equations for the roots of the polynomial. One solution is r_1 = 1, r_2 = -1, and r_3 = -3.
Once we have the roots of the polynomial, we can factor it by grouping. The roots of the polynomial are the zeros of the polynomial, so they divide the polynomial evenly. We can use this fact to write the polynomial as a product of two binomials:
(x-r_1)(x-r_2)(x-r_3) In this case, the roots of the polynomial are 1, -1, and -3. Therefore, the polynomial can be factored as follows:
(x-1)(x+1)(x-3) We can also factor the polynomial by grouping. The first two terms of the polynomial are (x^3-3x^2), and the last two terms are (-x+3). The greatest common factor of the first two terms is x^2, and the greatest common factor of the last two terms is -1. Therefore, we can factor the polynomial as follows:
(x^2)(x-3) - (-1)(-x+3) This simplifies to:
(x^2)(x-3) + (x-3) We can then factor out (x-3) to get:
(x-3)(x^2+1) This is the same factorization that we obtained using the sum-product pattern.
Related Questions
- What are the roots of the polynomial 3x^3-3x^2-x+3?
- How can we factor the polynomial 3x^3-3x^2-x+3 using the sum-product pattern?
- How can we factor the polynomial 3x^3-3x^2-x+3 by grouping?
Answers
- The roots of the polynomial 3x^3-3x^2-x+3 are 1, -1, and -3.
- We can factor the polynomial 3x^3-3x^2-x+3 using the sum-product pattern by writing down the following equations:
r_1 + r_2 + r_3 = -3 r_1 r_2 r_3 = 1 Solving these equations, we get r_1 = 1, r_2 = -1, and r_3 = -3.
- We can factor the polynomial 3x^3-3x^2-x+3 by grouping by writing the polynomial as follows:
(x^3-3x^2) - (-x+3) The greatest common factor of the first two terms is x^2, and the greatest common factor of the last two terms is -1. Therefore, we can factor the polynomial as follows:
(x^2)(x-3) + (x-3) This simplifies to:
(x-3)(x^2+1) 