Question

Determine roots of the polynomial 2x + 2x^2 + x + 1. Explain how to determine potential rational roots.

Answer 1

The roots of the polynomial
`\(2x + 2x^2 + x + 1\)`are (x = -1) and
`\(x = -(1)/(2)\).`

Step-by-step explanation:

To determine the roots of the given polynomial, we first factorize it by grouping:

`\[2x + 2x^2 + x + 1 = 2x(x + 1) + 1(x + 1) = (2x + 1)(x + 1).\]`

Setting each factor to zero gives the potential roots:

`\[2x + 1 = 0 \implies x = -(1)/(2)\]`

and

`\[x + 1 = 0 \implies x = -1.\]`

Thus, the roots are (x = -1) and
`\(x = -(1)/(2)\).` These are the values for which the polynomial evaluates to zero. This is because when a product of factors equals zero, at least one of the factors must be zero.

To explain how to determine potential rational roots, we can use the Rational Root Theorem. According to this theorem, the potential roots are all possible ratios of factors of the constant term (1 in this case) to factors of the leading coefficient (2 in this case). In our polynomial, the potential rational roots are
`\(\pm 1\) and \(\pm (1)/(2)\),` which aligns with the roots we found earlier. This theorem provides a systematic way to identify the potential rational roots without having to test all possible values, significantly simplifying the root-finding process.