Question

Use synthetic division to solve (3x^4+6x^3+2x^2+9x+10) divided by (x+2) What is the quotient?

Please help.

Answer 1

Answer:

The quotient is $3x^3+2x+5$.

Step-by-step explanation:

The dividend is

$3x^4+6x^3+2x^2+9x+10$

The coefficient of dividend are 3,6,2,9 and the constant term is 10.

The divisor is (x+2), then divisor in the for of (x-c), therefore the value of c is -2.

Write -2 and the coefficient in the top row.

Now write first coefficient as it is in bottom row.

Multiply -2 and 3 are write the result in middle row. Then add 6 and -6.

Apply same process as shown below.

The bottom row show the coefficient and the last term of the bottom row represents the remainder.

The degree of the quotient is one less than the degree of the polynomial. Therefore the degree of quotient is 3. The quotient is

$3x^3+0x^2+2x+5$

$3x^3+2x+5$

The remainder is 0.

$3x^4+6x^3+2x^2+9x+10=(x+2)(3x^3+2x+5)$

Answer 2

The quotient is of the division of $3{x^4} + 6{x^3} + 2{x^2} + 9x + 10$ and $x +2$ is $\boxed{3{x^3} + 2x + 5}$ and the remainder is $\boxed0.$

Further explanation:

The numerator of the division is $3{x^4} + 6{x^3} + 2{x^2} + 9x + 10$ and the denominator is $x +2.$

Solve the given polynomial $P\left( x \right) = 3{x^4} + 6{x^3} + 2{x^2} + 9x + 10$ to obtain the quotient by the use of synthetic division.

Now obtain the value of x from the denominator.

$\begin{aligned}x + 2&= 0\\x&=- 2\\\end{aligned}$

Divide the coefficients of the polynomial by $- 2$, to check whether $- 2$is a zero of the polynomial.

$\begin{aligned}2\left| \!{\nderline {\,{3\,\,\,\,\,\,\,\,\,\,6\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,9\,\,\,\,\,\,\,\,\,\,\,\,\,10} \,}} \right.  \hfill \\\,\,\underline {\,\,\,\,\,\,\,\,\, - 6\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\, - 4\,\,\,\,\,\,\,\, - 10}\hfill\\\,\,\,\underline{\,\,\,3\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,5\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,}\hfill\\\end{aligned}$

The last entry of the synthetic division told about remainder and the last entry of the synthetic division is 0. Therefore, the remainder of the synthetic division is 0.

The entry in the last line of synthetic division gives us the coefficients of the quotient.

The new numbers are the coefficients of the quotient.

The quotient is of the division of $3{x^4} + 6{x^3} + 2{x^2} + 9x + 10$ and $x + 1$ is $\boxed{3{x^3} + 2x + 5}$ and the remainder is $\boxed0.$

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Synthetic Division

Keywords: division, synthetic division, long division method, coefficients, quotients, remainders, numerator, denominator, polynomial, zeros, degree.