factor completely using synthetic division. one zero is given. 1. f(x)=x³-8x³+19x-12;3        2. f(x)=x⁴+2x³-13x+10x;-5        3. f(x)= x³+6x²+11x+6;-2

Let's tackle these polynomials using synthetic division. We'll use the given zero to find a quotient, and then factor the quotient further if possible.

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### 1. \( f(x) = x^3 - 8x^2 + 19x - 12 \); Zero: \( 3 \)

Perform synthetic division by dividing the polynomial by \( (x - 3) \):

1. Write the coefficients: \( 1, -8, 19, -12 \).
2. Set up synthetic division with \( 3 \) as the divisor.

\[
\begin{array}{r|rrrr}
3 & 1 & -8 & 19 & -12 \\
  &   & 3  & -15 & 12 \\
\hline
  & 1 & -5 & 4  & 0 \\
\end{array}
\]

The quotient is \( x^2 - 5x + 4 \). Now, factor this quadratic:

\[
x^2 - 5x + 4 = (x - 1)(x - 4)
\]

So, the complete factorization of \( f(x) \) is:

\[
f(x) = (x - 3)(x - 1)(x - 4)
\]

---

### 2. \( f(x) = x^4 + 2x^3 - 13x^2 + 10x \); Zero: \( -5 \)

Perform synthetic division by dividing the polynomial by \( (x + 5) \):

1. Write the coefficients: \( 1, 2, -13, 10, 0 \) (don't forget the \( x^0 \) term, which has a coefficient of 0).
2. Set up synthetic division with \( -5 \) as the divisor.

\[
\begin{array}{r|rrrrr}
-5 & 1 & 2 & -13 & 10 & 0 \\
   &   & -5 & 15 & -10 & 0 \\
\hline
   & 1 & -3 & 2 & 0 & 0 \\
\end{array}
\]

The quotient is \( x^3 - 3x^2 + 2x \). Now, factor this further:

\[
x(x^2 - 3x + 2) = x(x - 1)(x - 2)
\]

So, the complete factorization of \( f(x) \) is:

\[
f(x) = (x + 5)x(x - 1)(x - 2)
\]

---

### 3. \( f(x) = x^3 + 6x^2 + 11x + 6 \); Zero: \( -2 \)

Perform synthetic division by dividing the polynomial by \( (x + 2) \):

1. Write the coefficients: \( 1, 6, 11, 6 \).
2. Set up synthetic division with \( -2 \) as the divisor.

\[
\begin{array}{r|rrrr}
-2 & 1 & 6 & 11 & 6 \\
   &   & -2 & -8 & -6 \\
\hline
   & 1 & 4 & 3 & 0 \\
\end{array}
\]

The quotient is \( x^2 + 4x + 3 \). Now, factor this quadratic:

\[
x^2 + 4x + 3 = (x + 3)(x + 1)
\]

So, the complete factorization of \( f(x) \) is:

\[
f(x) = (x + 2)(x + 3)(x + 1)
\]

---

### Summary of Factored Forms:

1. \( f(x) = (x - 3)(x - 1)(x - 4) \)
2. \( f(x) = (x + 5)x(x - 1)(x - 2) \)
3. \( f(x) = (x + 2)(x + 3)(x + 1) \)

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Would you like more details on any of these steps?

Here are 5 related questions to explore:
1. What is synthetic division, and why is it useful?
2. How do we check if a number is a zero of a polynomial?
3. Can synthetic division work with non-integer zeros?
4. What happens if the remainder isn't zero in synthetic division?
5. How do we factor complex polynomials with no rational zeros?

**Tip:** Always remember to verify your results by substituting the zeros back into the original polynomial.

Factor completely using synthetic division. One zero is given: 1. f(x)=x³-8x³+19x-12;3 2. f(x)=x⁴+2x³-13x+10x;-5 3. f(x)=x³+6x²+11x+6;-2

Learn how to factor polynomials using synthetic division when a zero is given. This step-by-step solution covers examples like f(x) = x³ - 8x² + 19x - 12 and f(x) = x³ + 6x² + 11x + 6, using synthetic division to find factors and applying the factor theorem. Ideal for high school students in Grades 10-12.

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