Questions: Divide using synthetic division. (3x^2+7x-5) ÷ (x+4)

Solution Steps

To divide a polynomial by a binomial using synthetic division, follow these steps:

1. Identify the coefficients of the dividend polynomial and the root of the divisor.
2. Set up the synthetic division table with the root of the divisor and the coefficients of the dividend.
3. Perform synthetic division by bringing down the leading coefficient, multiplying it by the root, and adding it to the next coefficient.
4. Continue this process until all coefficients have been processed.
5. The result will be the coefficients of the quotient polynomial, with the last number being the remainder.

The polynomial to be divided is \(3x^2 + 7x - 5\). The coefficients are:

• \(a_0 = 3\)
• \(a_1 = 7\)
• \(a_2 = -5\)

The divisor is \(x + 4\), which has a root of \(r = -4\).

We set up the synthetic division using the coefficients \([3, 7, -5]\) and the root \(-4\).

1. Bring down the leading coefficient: \(3\).
2. Multiply \(3\) by \(-4\) to get \(-12\) and add it to \(7\): \[ 7 + (-12) = -5 \]
3. Multiply \(-5\) by \(-4\) to get \(20\) and add it to \(-5\): \[ -5 + 20 = 15 \]

The synthetic division process yields the coefficients of the quotient polynomial and the remainder:

• Quotient: \(3x + -5\) (or \(3x - 5\))
• Remainder: \(15\)

Final Answer