Let p(x)=x^3+4x^2+x-6. Find a polynomial g(x) can be (remainder) r, such that p(x) can be expressed in the form p(x)=(x-2)g(x)+r
To find the polynomial g(x) and the remainder r, we can use polynomial long division. Here's how you can do it step by step:
Step 1: Write the dividend polynomial p(x) and the divisor polynomial (x - 2) in the standard form, arranged in descending order of exponents:
p(x) = x^3 + 4x^2 + x - 6
(x - 2)
Step 2: Divide the first term of the dividend by the first term of the divisor to get the quotient's leading term:
( x^3 )/( x ) = x^2
Step 3: Multiply the divisor (x - 2) by the obtained quotient term (x^2) and subtract it from the dividend. Write down the result:
x^2
(x - 2) / x^3 + 4x^2 + x - 6
- ( x^3 - 2x^2 )
-----------------
6x^2 + x - 6
Step 4: Repeat steps 2 and 3 with the new dividend (6x^2 + x - 6) and the same divisor (x - 2):
x^2 + 6x
(x - 2) / 6x^2 + x - 6
- ( 6x^2 - 12x )
-----------------
13x - 6
Step 5: Repeat steps 2 and 3 once again:
x^2 + 6x + 13
(x - 2) / 13x - 6
- ( 13x - 26 )
-------------
20
Step 6: We have now reached a polynomial (20) that cannot be divided further. This is the remainder r.
Therefore, the polynomial g(x) is obtained by combining the quotient terms: g(x) = x^2 + 6x + 13, and the remainder r is 20.
Hence, the polynomial p(x) can be expressed in the form p(x) = (x - 2)g(x) + r as requested.