We read your photo as:

Understand the Factor Theorem.

April 7, 2026Mathematics

ScanSolve AI Answer

Answer

\frac{7}{2}

4 steps

Understand the Factor Theorem.
The Factor Theorem states that if $(x-c)$ is a factor of a polynomial $P(x)$ , then $P(c) = 0$ .
Identify the polynomial and the factor.
The given polynomial is $P(x) = 2x^3+4x^2+px+7$ .
The given factor is $(x+2)$ .
Comparing $(x+2)$ with $(x-c)$ , we find that $c = -2$ .
Apply the Factor Theorem.
Since $(x+2)$ is a factor of $P(x)$ , we must have $P(-2) = 0$ .
Substitute $x = -2$ into the polynomial:
$P(-2) = 2(-2)^3 + 4(-2)^2 + p(-2) + 7$
Simplify the expression and solve for $p$ .
$2(-8) + 4(4) - 2p + 7 = 0$ $-16 + 16 - 2p + 7 = 0$ $0 - 2p + 7 = 0$ $-2p + 7 = 0$ $-2p = -7$ $p = \frac{-7}{-2}$ $p = \frac{7}{2}$
The value of $p$ is $\boxed{\frac{7}{2}}$ .