Math Problem Statement

Use synthetic division to determine which of the following is a factor of 3x^3 - 10x^2 + 3x + 10.

Solution

To solve this problem using synthetic division, we need to divide the polynomial 3x310x2+3x+103x^3 - 10x^2 + 3x + 10 by each of the given factors (corresponding to x+2x + 2, x3x - 3, x+3x + 3, and x2x - 2) and determine which one yields a remainder of zero. A remainder of zero means the divisor is a factor.

Step-by-Step Synthetic Division:

We'll perform synthetic division for each option.

1. x+2x + 2 (which means we divide by x=2x = -2):

The coefficients of the polynomial 3x310x2+3x+103x^3 - 10x^2 + 3x + 10 are 3,10,3,103, -10, 3, 10.

Set up synthetic division with -2 as the divisor:

-2 & 3 & -10 & 3 & 10 \\ & & -6 & 32 & -70 \\ \hline & 3 & -16 & 35 & -60 \\ \end{array}$$ The remainder is $$-60$$, so $$x + 2$$ is **not** a factor. #### 2. $$x - 3$$ (which means we divide by $$x = 3$$): Set up synthetic division with 3 as the divisor: $$\begin{array}{r|rrrr} 3 & 3 & -10 & 3 & 10 \\ & & 9 & -3 & 0 \\ \hline & 3 & -1 & 0 & 10 \\ \end{array}$$ The remainder is 10, so $$x - 3$$ is **not** a factor. #### 3. $$x + 3$$ (which means we divide by $$x = -3$$): Set up synthetic division with -3 as the divisor: $$\begin{array}{r|rrrr} -3 & 3 & -10 & 3 & 10 \\ & & -9 & 57 & -180 \\ \hline & 3 & -19 & 60 & -170 \\ \end{array}$$ The remainder is $$-170$$, so $$x + 3$$ is **not** a factor. #### 4. $$x - 2$$ (which means we divide by $$x = 2$$): Set up synthetic division with 2 as the divisor: $$\begin{array}{r|rrrr} 2 & 3 & -10 & 3 & 10 \\ & & 6 & -8 & -10 \\ \hline & 3 & -4 & -5 & 0 \\ \end{array}$$ The remainder is 0, so $$x - 2$$ **is** a factor of the polynomial. ### Conclusion: The correct answer is **(d) $$x - 2$$**. --- Would you like further details on synthetic division, or do you have any questions about the process? Here are 5 related questions for further practice: 1. What happens if the remainder in synthetic division is not zero? 2. How do you factor a cubic polynomial once you know one factor? 3. Can synthetic division be used for polynomials of any degree? 4. What is the difference between synthetic division and long division for polynomials? 5. How can you verify the result of synthetic division by multiplication? **Tip:** Always verify your synthetic division by multiplying the quotient with the divisor to ensure the original polynomial is obtained!

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Synthetic Division

Formulas

Synthetic Division Procedure

Theorems

Factor Theorem

Suitable Grade Level

Grades 9-12