Is the given binomial a factor of the given polynomial? if so, write the polynomial as a product of two factors. polynomial: \( P(x)=2 x^{3}+5 x^{2}-8 x-20 ; \) binomial: \( 2 x+5 \)

To determine if the given binomial is a factor of the given polynomial, we can use polynomial long division. Given polynomial: \( P(x) = 2x^3 + 5x^2 - 8x - 20 \) Given binomial: \( 2x + 5 \) We will perform polynomial long division to divide the polynomial by the binomial and check if the remainder is zero. If the remainder is zero, then the binomial is a factor of the polynomial. Let's perform the polynomial long division: Divide the polynomials by following steps: - step0: Using Long Division to Divide Polynomials: \(\left(2x^{3}+5x^{2}-8x-20\right)\div \left(2x+5\right)\) - step1: Calculate: \(\left(x^{2}-4\right)\text{ }\textrm{remainder}\text{ }0\) The result of the polynomial long division is \( (x^2 - 4) \) with a remainder of 0. This means that the binomial \( 2x + 5 \) is a factor of the polynomial \( P(x) = 2x^3 + 5x^2 - 8x - 20 \). Therefore, the polynomial can be written as a product of two factors: \[ P(x) = (2x + 5)(x^2 - 4) \]