Property 13
Example
| \(P(x)\) | \(x^2-4\) |
| _ |
| \(3x-5\) |
| \(P(x)\) | \(x-2\) |
| _ |
| \(K\) |
\(\Rightarrow K=?\)
Answer
| \(P(x)\) | \(x^2-4\) |
| _ | \(B_1(x)\) |
| \(3x-5\) |
\(\Rightarrow P(x)=(x^2-4)\cdot B_1(x)+3x-5\)
\(P(2)=3\cdot2-5=1\quad\text{(I)}\)
| \(P(x)\) | \(x-2\) |
| _ | \(B_2(x)\) |
| \(K\) |
\(\Rightarrow P(x)=(x-2)\cdot B_2(x)+K\)
\(P(2)=K\quad\text{(II)}\)
\(\text{(I)}=\text{(II)}\Rightarrow K=1\)
1.
| \(P(x)\) | \(x^3-8\) |
| _ |
| \(x^2+x-2\) |
| \(P(x)\) | \(x-2\) |
| _ |
| \(K\) |
\(\Rightarrow K=?\)
2.
| \(P(x)\) | \(x^2-3x+2\) |
| _ |
| \(x+4\) |
| \(P(x)\) | \(x-1\) |
| _ |
| \(K\) |
\(\Rightarrow K=?\)
3.
| \(P(x)\) | \((x-4)^2\) |
| _ |
| \(6-8x\) |
| \(P(x)\) | \(x-4\) |
| _ |
| \(K\) |
\(\Rightarrow K=?\)
4.
| \(P(x)\) | \(x^2+5x\) |
| _ |
| \(7x-15\) |
| \(P(x)\) | \(x+5\) |
| _ |
| \(K\) |
\(\Rightarrow K=?\)
5.
| \(P(x)\) | \(x^2+6x\) |
| _ |
| \(x-5\) |
| \(P(x)\) | \(x+6\) |
| _ |
| \(K\) |
\(\Rightarrow K=?\)
6.
| \(P(x)\) | \(x^3-27\) |
| _ |
| \(x^2-5x+6\) |
| \(P(x)\) | \(x^2+3x+9\) |
| _ |
| \(K\) |
\(\Rightarrow K=?\)
7.
| \(P(x)\) | \(x^2-x-1\) |
| _ |
| \(2x-1\) |
| \(P^2(x)\) | \(x^2-x-1\) |
| _ |
| \(K(x)\) |
\(\Rightarrow K(x)=?\)
Property 14
Horner Method
| 2. | 1. |
| 3. |
By using Horner method, a table above can be done to divide a P(x) polynomial to “x − a”.
- Factors of P(x) polynomial is written in 1st region according to the decreasing exponents of x.
- x − a = 0 ⇒ x = a (value of “a” is written in 2nd region)
- The leading coefficient is put down to 3rd region.
- The value of “a” is multiplied by the leading coefficient, and it is added to the next coefficient. This operation is applied to all coefficients.
- The final numbers gives us the remainder value and the others give the division of polynomial factors.
Example
\(P(x)=x^3-2x^2+x+4\)
Find the remainder and division of P(x) = x³ − 2x² + x + 4 polynomial divided by x − 3.
Answer
| \(3\) | \(1\) | \(-2\) | \(1\) | \(4\) |
| \(3\cdot1\) | \(3\cdot1\) | \(3\cdot4\) | ||
| \(1\) | \(1\) | \(4\) | \(16\) | |
| (Factors of polynomial division) | (Remainder) | |||
Polynomial division
\(1\cdot x^2+1\cdot x+4\cdot x^0=x^2+x+4\)
Polynomial remainder
\(16\)
1.
| \(2x^3+3x^2-5x+7\) | \(x-1\) |
| _ | \(B(x)\) |
| \(K(x)\) |
\(\Rightarrow B(x)=?\)
\(\Rightarrow K(x)=?\)
2.
| \(x^3+x+1\) | \(x+1\) |
| _ | \(B(x)\) |
| \(K(x)\) |
\(\Rightarrow B(x)=?\)
\(\Rightarrow K(x)=?\)
3.
| \(2x^4-x^3+x^2+x+1\) | \(x-2\) |
| _ | \(B(x)\) |
| \(K(x)\) |
\(\Rightarrow B(x)=?\)
\(\Rightarrow K(x)=?\)
4.
| \(2x^3-x^2+3\) | \(x+3\) |
| _ | \(B(x)\) |
| \(K(x)\) |
\(\Rightarrow B(x)=?\)
\(\Rightarrow K(x)=?\)
5.
| \(2x^3-3x^2+7x+3\) | \(x-2\) |
| _ | \(B(x)\) |
| \(K(x)\) |
\(\Rightarrow B(x)=?\)
\(\Rightarrow K(x)=?\)
6.
| \(3x^4-4x^3-2x^2+5x+5\) | \(x-1\) |
| _ | \(B(x)\) |
| \(K(x)\) |
\(\Rightarrow B(x)=?\)
\(\Rightarrow K(x)=?\)
7.
| \(2x^3-11x^2+6\) | \(x-3\) |
| _ | \(B(x)\) |
| \(K(x)\) |
\(\Rightarrow B(x)=?\)
\(\Rightarrow K(x)=?\)
8.
| \(6x^4-6x^2+2\) | \(x-2\) |
| _ | \(B(x)\) |
| \(K(x)\) |
\(\Rightarrow B(x)=?\)
\(\Rightarrow K(x)=?\)
9.
| \(2x^4+x^3+4x+1\) | \(x-2\) |
| _ | \(B(x)\) |
| \(K(x)\) |
\(\Rightarrow B(x)=?\)
\(\Rightarrow K(x)=?\)
10.
| \(-x^3+3x^2-9\) | \((x-1)^2\) |
| _ | \(B(x)\) |
| \(K(x)\) |
\(\Rightarrow B(x)=?\)
\(\Rightarrow K(x)=?\)
11.
| \(x^3-x^2+x+1\) | \((x-2)^2\) |
| _ | \(B(x)\) |
| \(K(x)\) |
\(\Rightarrow B(x)=?\)
\(\Rightarrow K(x)=?\)
12.
| \(-4x^4+4x-5\) | \((x-1)\cdot(x+2)\) |
| _ | \(B(x)\) |
| \(K(x)\) |
\(\Rightarrow B(x)=?\)
\(\Rightarrow K(x)=?\)