Question

Select the polynomial that is divisible by (x – 1).

1.)p(x) = 3.23 + 2x²
2.)p(x) = 2x³ – 3x² + 2a + 1
3.)p(x) = 5x³ – 4x²
4.) p(x) = x³ + 2x² + 3x + 2

Answer 1

None of the given polynomials are divisible by (x - 1).

Step-by-step explanation:

In order for a polynomial to be divisible by (x - 1), the remainder when we substitute x = 1 into the polynomial should be 0. Let's check each polynomial:

1. p(x) = 3x² + 2
2. p(x) = 2x³ - 3x² + 2a + 1
3. p(x) = 5x³ - 4x² + 3
4. p(x) = x³ + 2x² + 3x + 2

When we substitute x = 1 into the first polynomial, we get 3(1)² + 2 = 5, which is not 0. Therefore, the first polynomial is not divisible by (x - 1).

When we substitute x = 1 into the second polynomial, we get 2(1)³ - 3(1)² + 2a + 1 = 2 - 3 + 2a + 1 = 2a. Since 2a can be any value, the second polynomial is not always divisible by (x - 1).

When we substitute x = 1 into the third polynomial, we get 5(1)³ - 4(1)² + 3 = 5 - 4 + 3 = 4. Therefore, the third polynomial is not divisible by (x - 1).

When we substitute x = 1 into the fourth polynomial, we get 1³ + 2(1)² + 3(1) + 2 = 1 + 2 + 3 + 2 = 8, which is not 0. Therefore, the fourth polynomial is not divisible by (x - 1).

Therefore, none of the given polynomials are divisible by (x - 1).