Question

use synthetic division to tell if the binomial is a factor of the polynomial.

1. $(3x^3 + 9x^2 + 4x + 12) \div (x + 3)$

1. $(3x^4 - 10x^3 - 24x^2 - 6x + 5) \div (x - 1)$

1. $(5x^4 - 46x^3 + 84x^2 - 50x + 7) \div (x - 7)$

Explanation:

Problem 3: $(3x^3 + 9x^2 + 4x + 12) \div (x + 3)$

Step1: Identify root for divisor

Root of $x+3=0$ is $x=-3$

Step2: Set up synthetic division

Coefficients: $3, 9, 4, 12$; root: $-3$

$$\begin{array}{r|rrrr} -3 & 3 & 9 & 4 & 12 \\ & & -9 & 0 & -12 \\ \hline & 3 & 0 & 4 & 0 \end{array}$$

Step3: Check remainder

Final value (remainder) is $0$

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Problem 4: $(3x^4 - 10x^3 - 24x^2 - 6x + 5) \div (x - 1)$

Step1: Identify root for divisor

Root of $x-1=0$ is $x=1$

Step2: Set up synthetic division

Coefficients: $3, -10, -24, -6, 5$; root: $1$

$$\begin{array}{r|rrrrr} 1 & 3 & -10 & -24 & -6 & 5 \\ & & 3 & -7 & -31 & -37 \\ \hline & 3 & -7 & -31 & -37 & -32 \end{array}$$

Step3: Check remainder

Final value (remainder) is $-32$

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Problem 5: $(5x^4 - 46x^3 + 84x^2 - 50x + 7) \div (x - 7)$

Step1: Identify root for divisor

Root of $x-7=0$ is $x=7$

Step2: Set up synthetic division

Coefficients: $5, -46, 84, -50, 7$; root: $7$

$$\begin{array}{r|rrrrr} 7 & 5 & -46 & 84 & -50 & 7 \\ & & 35 & -77 & 49 & -7 \\ \hline & 5 & -11 & 7 & -1 & 0 \end{array}$$

Step3: Check remainder

Final value (remainder) is $0$

Answer:

1. The binomial $x+3$ is a factor of $3x^3 + 9x^2 + 4x + 12$ (remainder = 0)
2. The binomial $x-1$ is NOT a factor of $3x^4 - 10x^3 - 24x^2 - 6x + 5$ (remainder = -32)
3. The binomial $x-7$ is a factor of $5x^4 - 46x^3 + 84x^2 - 50x + 7$ (remainder = 0)