To find the other factor of the polynomial 6x² + 15, given that one factor is 2x + 3, we will use polynomial division.
First, let’s verify if 2x + 3 is indeed a factor of the polynomial 6x² + 15. We will do this by performing polynomial long division.
Step 1: Divide the leading terms
We start with the leading term of the dividend 6x² and divide it by the leading term of the divisor 2x:
6x² ÷ 2x = 3x.
Step 2: Multiply and subtract
Next, we multiply the entire divisor (2x + 3) by 3x:
(2x + 3) * 3x = 6x² + 9x.
Now, we subtract this result from the original polynomial:
(6x² + 15) – (6x² + 9x) = 15 – 9x.
Step 3: Divide again
Now we have a new polynomial of -9x + 15. We take the leading term -9x and divide it by 2x:
-9x ÷ 2x = - rac{9}{2}.
Step 4: Multiply and subtract again
Now we multiply the divisor by - rac{9}{2}:
(2x + 3) * - rac{9}{2} = -9x – rac{27}{2}.
Subtract this from -9x + 15:
(-9x + 15) – (-9x – rac{27}{2}) = 15 + rac{27}{2}.
This gives us:
15 + rac{27}{2} = rac{30}{2} + rac{27}{2} = rac{57}{2}.
Conclusion
At this point, we have determined the quotient from our division, which is 3x – rac{9}{2}. Therefore, the other factor, alongside the given factor of 2x + 3, can be expressed as:
6x² + 15 = (2x + 3)(3x – rac{9}{2}).
Hence, the other factor of the polynomial 6x² + 15, when 2x + 3 is provided as a factor, is 3x – rac{9}{2}.