given that (x+2)s a factor of 2x^3+ 6x^2 + bx - 5 find the remainder when the expression is divided by (2x - 1)
Well let's factor it then to get
(x+2)(2x^2 + 2x - 5/2)
I'll let you figure out b and then do the division to find the remainder.
To find the remainder when the expression is divided by (2x - 1), we can use the polynomial division method. Here's how you can proceed:
1. Start by writing the dividend, which is 2x^3 + 6x^2 + bx - 5, and the divisor, which is 2x - 1.
2. Perform the long division by dividing the first term of the dividend (2x^3) by the first term of the divisor (2x). This gives you x^2.
3. Multiply the divisor (2x - 1) by the quotient obtained in step 2 (x^2). This gives you 2x^3 - x^2.
4. Subtract the product obtained in step 3 (2x^3 - x^2) from the original dividend (2x^3 + 6x^2 + bx - 5). This gives you 7x^2 + bx - 5.
5. Repeat steps 2-4 with the new dividend (7x^2 + bx - 5) until you cannot divide further.
At this point, we know that (x + 2) is a factor, which means that when we divide the expression by (2x - 1), we should obtain a quotient of (2x^2 + 2x - 5/2), as you correctly mentioned.
Now, to find the remainder, let's continue the polynomial division:
6. Divide the first term of the new dividend (7x^2) by the first term of the divisor (2x). This gives you 3.5x.
7. Multiply the divisor (2x - 1) by the quotient obtained in step 6 (3.5x). This gives you 7x^2 - 3.5x.
8. Subtract the product obtained in step 7 (7x^2 - 3.5x) from the new dividend (7x^2 + bx - 5). This gives you (b - 3.5)x - 5.
9. Set the resulting expression (b - 3.5)x - 5 equal to zero, since it is the remainder.
Therefore, the remainder when the expression 2x^3 + 6x^2 + bx - 5 is divided by (2x - 1) is (b - 3.5)x - 5.