The Long and Short of It
1. Review dividing polynomials by a monomial.
2. Master the universal method: Polynomial Long Division.
3. Learn the shortcut method: Synthetic Division.
Remember the Quotient Rule: When dividing, you subtract exponents.
\(\frac{x^a}{x^b} = x^{a-b}\)
\(\frac{x^7}{x^3} = x^{7-3} = x^4\)
Split the problem into separate fractions and simplify each one.
\(\frac{8x^4 + 4x^3 - 2x^2}{2x^2}\)
becomes
\(\frac{8x^4}{2x^2} + \frac{4x^3}{2x^2} - \frac{2x^2}{2x^2}\)
\( = 4x^2 + 2x - 1\)
Let's solve \(\frac{8x^4 + 4x^3 - 2x^2}{2x^2}\)
Step 1: Split the polynomial into separate fractions, with each term over the monomial divisor.
\(\frac{8x^4}{2x^2} + \frac{4x^3}{2x^2} - \frac{2x^2}{2x^2}\)
Step 2: Simplify each fraction. Divide the coefficients and subtract the exponents.
\(( \frac{8}{2}x^{4-2} ) + ( \frac{4}{2}x^{3-2} ) - ( \frac{2}{2}x^{2-2} )\)
Step 3: Write the final simplified expression.
\(4x^2 + 2x - 1\)
Polynomial Long Division works for any division problem. The process is the same as number division!
With Numbers...
With Polynomials...
Divide, Multiply, Subtract, Bring Down
Let's solve \( (x^2 + 7x + 10) \div (x+2) \)
① Divide \(x^2\) by \(x\) to get \(x\).
② Multiply \(x(x+2)\) to get \(x^2+2x\).
③ Subtract and Bring Down the 10.
④ Repeat: Divide \(5x\) by \(x\) to get 5.
⑤ Multiply \(5(x+2)\) and subtract.
Answer: \(x+5\)
Synthetic Division is faster, but only works when the divisor is \( (x-k) \) or \( (x+k) \).
\((x^2+2x-1) \div (x-3)\)
\((3x^3-5) \div (x+1)\)
\((x^3+1) \div (x^2+1)\)
\((2x^2-x-1) \div (3x-5)\)
Let's solve \( (x^2 + 7x + 10) \div (x+2) \)
① Set up: Use -2 from \((x+2)\) and the coefficients.
② Bring down the first coefficient (1).
③ Multiply & Add: \(-2 \times 1 = -2\). Then \(7 + (-2) = 5\).
④ Multiply & Add: \(-2 \times 5 = -10\). Then \(10 + (-10) = 0\).
⑤ The result is \(1x+5\) with a remainder of 0.
Answer: \(x+5\)
Solving \((2x^3 - 9x^2 + 10x - 7) \div (x-3)\)
Answer: \(2x^2 - 3x + 1 - \frac{4}{x-3}\)
Choose the best method for each. Good luck!
Problem 1
\((k^2 - 7k + 10) \div (k-5)\)
\(k-2\)
Problem 2
\((2x^2 + 10x + 8) \div (x+4)\)
\(2x+2\)
Simplify the following expression using your preferred method:
The answer is \(v^2+v\)