Sure! To find the other factor of the polynomial [tex]\( 6x^2 - 17x - 14 \)[/tex] given that one factor is [tex]\( 2x - 7 \)[/tex], we can use polynomial long division.
Here’s a step-by-step solution:
1. Setup the Division: Arrange the polynomial and the factor in the long division format:
[tex]\[
\text{Dividend: } 6x^2 - 17x - 14
\][/tex]
[tex]\[
\text{Divisor: } 2x - 7
\][/tex]
2. First Division Step: Divide the leading term of the dividend by the leading term of the divisor.
[tex]\[
\frac{6x^2}{2x} = 3x
\][/tex]
The first term of the quotient is [tex]\(3x\)[/tex].
3. Multiply and Subtract: Multiply [tex]\(3x\)[/tex] by the entire divisor [tex]\(2x - 7\)[/tex]:
[tex]\[
3x \cdot (2x - 7) = 6x^2 - 21x
\][/tex]
Subtract this result from the original polynomial:
[tex]\[
(6x^2 - 17x - 14) - (6x^2 - 21x) = 4x - 14
\][/tex]
4. Next Division Step: Bring down the next term, which is already included. Now, divide the leading term of the new polynomial by the leading term of the divisor:
[tex]\[
\frac{4x}{2x} = 2
\][/tex]
The next term of the quotient is [tex]\(2\)[/tex].
5. Multiply and Subtract Again: Multiply [tex]\(2\)[/tex] by the entire divisor [tex]\(2x - 7\)[/tex]:
[tex]\[
2 \cdot (2x - 7) = 4x - 14
\][/tex]
Subtract this result from the current polynomial:
[tex]\[
(4x - 14) - (4x - 14) = 0
\][/tex]
Since the remainder is [tex]\(0\)[/tex], this confirms that [tex]\(2x - 7\)[/tex] is indeed a factor.
6. Construct the Quotient: The quotient found from this division is [tex]\(3x + 2\)[/tex].
Therefore, the other factor of the polynomial [tex]\(6x^2 - 17x - 14\)[/tex] is [tex]\(3x + 2\)[/tex].
So, the factorization of [tex]\(6x^2 - 17x - 14\)[/tex] is:
[tex]\[
6x^2 - 17x - 14 = (2x - 7)(3x + 2)
\][/tex]
