For the polynomial below, -3 is a zero.

Express as a product of linear factors.

Ask by Mcdonald Davison. in the United States

Mar 25,2025

Question

For the polynomial below, -3 is a zero. $$ f(x)=x^{3}+6 x^{2}+11 x+6 $$ Express $$ f(x) $$ as a product of linear factors. $$ f(x)=\square $$

UpStudy AI · Accepted Answer

We are given the polynomial
$$
f(x) = x^3 + 6x^2 + 11x + 6.
$$
Since $$-3$$ is a zero, by the Factor Theorem, $$(x + 3)$$ is a factor of $$f(x)$$.

1. **Divide $$f(x)$$ by $$(x+3)$$:**

We can use synthetic division with $$-3$$:

$$
   \begin{array}{cccc}
   -3 & \quad | & 1 & \quad 6 & \quad 11 & \quad 6 \
      &         &   & -3    & \quad -9 & \quad -6 \
      \hline
      &         & 1 & \quad 3 & \quad 2 & \quad 0 \
   \end{array}
   $$

The quotient is the quadratic polynomial
   $$
   x^2 + 3x + 2.
   $$

2. **Factor the quadratic:**

Factor $$x^2 + 3x + 2$$ by finding two numbers whose product is $$2$$ and sum is $$3$$. These numbers are $$1$$ and $$2$$. Hence,
   $$
   x^2 + 3x + 2 = (x + 1)(x + 2).
   $$

3. **Express $$f(x)$$ as a product of linear factors:**

Combine the factors:
   $$
   f(x) = (x + 3)(x + 1)(x + 2).
   $$

The final factored form is:
$$
f(x) = (x+1)(x+2)(x+3).
$$
$$ f(x) = (x + 1)(x + 2)(x + 3) $$

For the polynomial below, -3 is a zero.

Express as a product of linear factors.

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For the polynomial below, -3 is a zero. Express as a product of linear factors.