Given the polynomial 2x3 − 5x2 + 6x − 15, rewrite the polynomial as a product of binomials
To rewrite the polynomial as a product of binomials, we need to factorize it.
By applying synthetic division, we find that one of the roots of the polynomial is x = 3.
So, we can divide the polynomial (2x^3 - 5x^2 + 6x - 15) by (x - 3) using synthetic division to get:
2 -5 6 -15
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3 | 2 -5 6 -15
- 6 -3 9
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2 1 3 -6
The quotient is 2x^2 + x + 3, and the remainder is -6.
Therefore, the polynomial 2x^3 - 5x^2 + 6x - 15 can be rewritten as the product of binomials as:
2x^3 - 5x^2 + 6x - 15 = (x - 3)(2x^2 + x + 3)
To rewrite the polynomial 2x^3 − 5x^2 + 6x − 15 as a product of binomials, we can use a process called factoring.
First, we look for any common factors in the polynomial. Here, we can see that all the coefficients are divisible by 1, so there are no common factors to pull out.
Next, we look for any potential rational roots using the Rational Root Theorem. The potential rational roots for this polynomial are ±1, ±3, ±5, ±15.
By trying each potential root using synthetic division or evaluating the polynomial at each value, we find that x = 1 is a root of the polynomial.
Using synthetic division with x = 1:
1 | 2 -5 6 -15
| 2 -3 3
| 2 -3 3 -12
The result of the synthetic division is 2x^2 - 3x + 3 with a remainder of -12.
So, we can rewrite the polynomial as:
2x^3 − 5x^2 + 6x − 15 = (x - 1)(2x^2 - 3x + 3) - 12
The polynomial 2x^3 - 5x^2 + 6x - 15 can be factored as a product of binomials as (x - 1)(2x^2 - 3x + 3) - 12.