Simplify (6x2 + 7x − 5) / (2x − 1) by dividing out.

Question

ScanSolve AI Tutor · Accepted Answer

Here's the solution for Question Q3:

To simplify the expression $\frac{6x^2 + 7x - 5}{2x - 1}$ by dividing out, we can use polynomial long division.

Set up the polynomial long division.
$\begin{array}{c|cc cc} \multicolumn{2}{r}{3x} & +5 \\ \cline{2-5} 2x-1 & 6x^2 & +7x & -5 \\ \multicolumn{2}{r}{-(6x^2} & -3x) \\ \cline{2-3} \multicolumn{2}{r}{0} & 10x & -5 \\ \multicolumn{2}{r}{} & -(10x & -5) \\ \cline{3-4} \multicolumn{2}{r}{} & 0 & 0 \\ \end{array}$
Divide the first term of the dividend ( $6x^2$ ) by the first term of the divisor ( $2x$ ).
$\frac{6x^2}{2x} = 3x$
Write $3x$ as the first term of the quotient.
Multiply the divisor ( $2x - 1$ ) by $3x$ .
$3x(2x - 1) = 6x^2 - 3x$
Subtract this result from the dividend.
$(6x^2 + 7x) - (6x^2 - 3x) = 10x$
Bring down the next term, $-5$ . The new dividend is $10x - 5$ .
Divide the first term of the new dividend ($10x$) by the first term of the divisor ($2x$).
$\frac{10x}{2x} = 5$
Write 5 as the next term of the quotient.
Multiply the divisor ( $2x - 1$ ) by 5.
$5(2x - 1) = 10x - 5$
Subtract this result from the new dividend.
$(10x - 5) - (10x - 5) = 0$
The remainder is 0.

The quotient is $3x + 5$ .

Alternatively, by factoring the numerator:
We look for two numbers that multiply to $6 \times -5 = -30$ and add to 7. These numbers are 10 and $-3$ .
$6x^2 + 7x - 5 = 6x^2 + 10x - 3x - 5$ $= 2x(3x + 5) - 1(3x + 5)$ $= (2x - 1)(3x + 5)$
Now, substitute this back into the fraction:
$\frac{(2x - 1)(3x + 5)}{2x - 1}$
Assuming $2x - 1 \neq 0$ , we can cancel out the common factor $(2x - 1)$ .
$= 3x + 5$
The simplified expression is $\boxed{\text{}3x + 5\text{}}$

Simplify (6x2 + 7x - 5) / (2x - 1) by dividing out.