Solve each problem. Is x+1 a factor of ƒ(x)=x^3+2x^2+3x+2?

Verified step by step guidance

1. Step 1: Use the Factor Theorem, which states that $x+1$ is a factor of $f(x)$ if and only if $f(-1)=0$.
2. Step 2: Substitute $x=-1$ into the polynomial $f(x)=x^3+2x^2+3x+2$.
3. Step 3: Calculate $f(-1)=(-1{)}^3+2(-1{)}^2+3(-1)+2$.
4. Step 4: Simplify the expression: $(-1{)}^3=-1$, $2(-1{)}^2=2$, $3(-1)=-3$, and $+2$.
5. Step 5: Add the results: $-1+2-3+2$ and check if the sum equals zero to determine if $x+1$ is a factor.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, ƒ(x) = x^3 + 2x^2 + 3x + 2 is a polynomial of degree 3. Understanding polynomial functions is essential for analyzing their behavior, including factors and roots.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors, which can be simpler polynomials. To determine if x + 1 is a factor of ƒ(x), one can use polynomial long division or synthetic division. If the remainder is zero, then x + 1 is indeed a factor.

Remainder Theorem

The Remainder Theorem states that for a polynomial f(x), if you divide it by (x - c), the remainder of this division is f(c). This theorem can be used to quickly check if x + 1 is a factor of ƒ(x) by evaluating ƒ(-1). If ƒ(-1) equals zero, then x + 1 is a factor.