a(x)= x(cubed)-5x(squared) + 7x + 3 is divided by b(x) = x - 2
*******_x^2__- 3_x__+_1________
(x-2) | x^3 - 5 x^2 + 7 x + 3
********x^3 - 2 x^2
************------------------
************- 3 x^2 + 7 x + 3
************- 3 x^2 + 6 x
************------------------
*********************** x + 3
*********************** x - 2
***********************-------
*********************** R = 5
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To find the result of dividing a polynomial by another polynomial, you can use polynomial division. This process is similar to long division with numbers.
Let's go step-by-step to divide the polynomial a(x) by b(x).
Step 1: Write the polynomials in decreasing order of powers.
a(x) = x^3 - 5x^2 + 7x + 3
b(x) = x - 2
Step 2: Divide the highest power term of a(x) by the highest power term of b(x).
In this case, (x^3) divided by (x) is x^2. Write this as the first term of the quotient.
Quotient = x^2
Step 3: Multiply the divisor (b(x)) by the quotient term obtained in step 2 and subtract it from the dividend (a(x)).
(x - 2) * (x^2) = x^3 - 2x^2
Subtracting this from a(x):
a(x) - (x^3 - 2x^2) = -3x^2 + 7x + 3
Step 4: Repeat steps 2 and 3 with the decreased polynomial (-3x^2 + 7x + 3) as the new dividend.
Dividing (-3x^2) by (x) gives -3x. Write this as the next term of the quotient.
Quotient = x^2 - 3x
Multiply the divisor (b(x)) by this new quotient term:
(x - 2) * (x^2 - 3x) = x^3 - 5x^2 + 6x
Subtracting this from the reduced polynomial:
(-3x^2 + 7x + 3) - (x^3 - 5x^2 + 6x) = 2x^2 + x + 3
Step 5: Repeat steps 2 and 3 with the new reduced polynomial (2x^2 + x + 3) as the new dividend.
Dividing (2x^2) by (x) gives 2x. Write this as the next term of the quotient.
Quotient = x^2 - 3x + 2
Multiply the divisor (b(x)) by this new quotient term:
(x - 2) * (x^2 - 3x + 2) = 2x^2 - 4x
Subtracting this from the reduced polynomial:
(2x^2 + x + 3) - (2x^2 - 4x) = 5x + 3
Step 6: The degree of the divisor b(x) is 1, and the degree of the remaining polynomial 5x + 3 is also 1. Since the degrees are the same, we can divide the coefficients.
Dividing (5x) by (x) gives 5. Write this as the next term of the quotient.
Quotient = x^2 - 3x + 2 + 5/(x - 2)
Finally, we have the quotient:
a(x) divided by b(x) = x^2 - 3x + 2 + 5/(x - 2)