%2F(x-7).jpeg "(x^3-13x^2+40x+18)/(x-7)")
Dividing Polynomials: A Step-by-Step Guide
In this article, we'll explore the process of dividing the polynomial (x^3 - 13x^2 + 40x + 18) by (x - 7).
Understanding Polynomial Division
Polynomial division is similar to long division with numbers. We aim to find a quotient polynomial and a remainder that satisfy the following equation:
(Dividend) = (Divisor) * (Quotient) + (Remainder)
In our case:
(x^3 - 13x^2 + 40x + 18) = (x - 7) * (Quotient) + (Remainder)
The Steps
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Set up the division:
___________ x - 7 | x^3 - 13x^2 + 40x + 18 -
Focus on the leading terms:
- Divide the leading term of the dividend (x^3) by the leading term of the divisor (x). This gives us x^2.
- Write x^2 above the dividend.
- Multiply the divisor (x - 7) by x^2: x^2 * (x - 7) = x^3 - 7x^2
- Subtract this result from the dividend:
x^2 x - 7 | x^3 - 13x^2 + 40x + 18 -(x^3 - 7x^2) ----------- -6x^2 + 40x -
Repeat the process:
- Bring down the next term of the dividend (40x).
- Divide the leading term of the new dividend (-6x^2) by the leading term of the divisor (x), which gives us -6x.
- Write -6x above the dividend.
- Multiply the divisor (x - 7) by -6x: -6x * (x - 7) = -6x^2 + 42x
- Subtract this result from the new dividend:
x^2 - 6x x - 7 | x^3 - 13x^2 + 40x + 18 -(x^3 - 7x^2) ----------- -6x^2 + 40x -(-6x^2 + 42x) ------------ -2x + 18 -
Continue the process:
- Bring down the last term of the dividend (18).
- Divide the leading term of the new dividend (-2x) by the leading term of the divisor (x), which gives us -2.
- Write -2 above the dividend.
- Multiply the divisor (x - 7) by -2: -2 * (x - 7) = -2x + 14
- Subtract this result from the new dividend:
x^2 - 6x - 2 x - 7 | x^3 - 13x^2 + 40x + 18 -(x^3 - 7x^2) ----------- -6x^2 + 40x -(-6x^2 + 42x) ------------ -2x + 18 -(-2x + 14) ---------- 4 -
The result:
The quotient is x^2 - 6x - 2 and the remainder is 4.
Therefore, we can write:
(x^3 - 13x^2 + 40x + 18) = (x - 7) * (x^2 - 6x - 2) + 4