Find the equation of the line of the below a) gradient 2 passes through (0,8) b) gradient 4 passes through…

Question

Find the equation of the line of the below a) gradient 2 passes through (0,8) b) gradient 4 passes through (0,3) 5. Divide a) (2x³4x² + 6x − 3) ÷ (x+4) b) (6x³ − 4x² + 3x + 2) ÷ (2x + 2)

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4. Find the equation of the line
The general equation of a line is $y = mx + c$ , where $m$ is the gradient (slope) and $c$ is the y-intercept.

a) gradient 2 passes through $(0, 8)$

Identify the gradient and the y-intercept.
The gradient is given as $m = 2$ .
The line passes through the point $(0, 8)$ . Since the x-coordinate is 0, this point is the y-intercept, so $c = 8$ .
Substitute the values into the equation $y = mx + c$ .
$y = 2x + 8$
The equation of the line is $\boxed{y = 2x + 8}$ .

b) gradient 4 passes through $(0, 3)$
Identify the gradient and the y-intercept.
The gradient is given as $m = 4$ .
The line passes through the point $(0, 3)$ . Since the x-coordinate is 0, this point is the y-intercept, so $c = 3$ .
Substitute the values into the equation $y = mx + c$ .
$y = 4x + 3$
The equation of the line is $\boxed{y = 4x + 3}$ .

5. Divide the polynomials

a) $(2x^3 4x^2 + 6x - 3) \div (x + 4)$
Assumption: The term $2x^3 4x^2$ is interpreted as $2x^3 + 4x^2$ . If it was intended as a product, the problem would be significantly more complex and usually written with a multiplication symbol.
We will perform polynomial long division for $(2x^3 + 4x^2 + 6x - 3) \div (x + 4)$ .
Divide $2x^3$ by $x$ .
$\begin{array}{c|cc cc} \multicolumn{2}{r}{2x^2} & & & \\ \cline{2-5} x+4 & 2x^3 & +4x^2 & +6x & -3 \\ \multicolumn{2}{r}{-(2x^3} & +8x^2) & & \\ \cline{2-3} \multicolumn{2}{r}{0} & -4x^2 & +6x & \\ \end{array}$
Divide $-4x^2$ by $x$ .
$\begin{array}{c|cc cc} \multicolumn{2}{r}{2x^2} & -4x & & \\ \cline{2-5} x+4 & 2x^3 & +4x^2 & +6x & -3 \\ \multicolumn{2}{r}{-(2x^3} & +8x^2) & & \\ \cline{2-3} \multicolumn{2}{r}{0} & -4x^2 & +6x & \\ \multicolumn{2}{r}{} & -(-4x^2 & -16x) & \\ \cline{3-4} \multicolumn{2}{r}{} & 0 & 22x & -3 \\ \end{array}$
Divide $22x$ by $x$ .
$\begin{array}{c|cc cc} \multicolumn{2}{r}{2x^2} & -4x & +22 & \\ \cline{2-5} x+4 & 2x^3 & +4x^2 & +6x & -3 \\ \multicolumn{2}{r}{-(2x^3} & +8x^2) & & \\ \cline{2-3} \multicolumn{2}{r}{0} & -4x^2 & +6x & \\ \multicolumn{2}{r}{} & -(-4x^2 & -16x) & \\ \cline{3-4} \multicolumn{2}{r}{} & 0 & 22x & -3 \\ \multicolumn{2}{r}{} & & -(22x & +88) \\ \cline{4-5} \multicolumn{2}{r}{} & & 0 & -91 \\ \end{array}$
The quotient is $2x^2 - 4x + 22$ and the remainder is $-91$ .
Therefore, $(2x^3 + 4x^2 + 6x - 3) \div (x + 4) = 2x^2 - 4x + 22 - \frac{91}{x+4}$ .

The result of the division is $\boxed{2x^2 - 4x + 22 - \frac{91}{x+4}}$ .

b) $(6x^3 - 4x^2 + 3x + 2) \div (2x + 2)$
We will perform polynomial long division for $(6x^3 - 4x^2 + 3x + 2) \div (2x + 2)$ .
Divide $6x^3$ by $2x$ .
$\begin{array}{c|cc cc} \multicolumn{2}{r}{3x^2} & & & \\ \cline{2-5} 2x+2 & 6x^3 & -4x^2 & +3x & +2 \\ \multicolumn{2}{r}{-(6x^3} & +6x^2) & & \\ \cline{2-3} \multicolumn{2}{r}{0} & -10x^2 & +3x & \\ \end{array}$
Divide $-10x^2$ by $2x$ .
$\begin{array}{c|cc cc} \multicolumn{2}{r}{3x^2} & -5x & & \\ \cline{2-5} 2x+2 & 6x^3 & -4x^2 & +3x & +2 \\ \multicolumn{2}{r}{-(6x^3} & +6x^2) & & \\ \cline{2-3} \multicolumn{2}{r}{0} & -10x^2 & +3x & \\ \multicolumn{2}{r}{} & -(-10x^2 & -10x) & \\ \cline{3-4} \multicolumn{2}{r}{} & 0 & 13x & +2 \\ \end{array}$
Divide $13x$ by $2x$.
$\begin{array}{c|cc cc} \multicolumn{2}{r}{3x^2} & -5x & +\frac{13}{2} & \\ \cline{2-5} 2x+2 & 6x^3 & -4x^2 & +3x & +2 \\ \multicolumn{2}{r}{-(6x^3} & +6x^2) & & \\ \cline{2-3} \multicolumn{2}{r}{0} & -10x^2 & +3x & \\ \multicolumn{2}{r}{} & -(-10x^2 & -10x) & \\ \cline{3-4} \multicolumn{2}{r}{} & 0 & 13x & +2 \\ \multicolumn{2}{r}{} & & -(13x & +13) \\ \cline{4-5} \multicolumn{2}{r}{} & & 0 & -11 \\ \end{array}$
The quotient is $3x^2 - 5x + \frac{13}{2}$ and the remainder is $-11$ .
Therefore, $(6x^3 - 4x^2 + 3x + 2) \div (2x + 2) = 3x^2 - 5x + \frac{13}{2} - \frac{11}{2x+2}$ .

The result of the division is $\boxed{3x^2 - 5x + \frac{13}{2} - \frac{11}{2x+2}}$ .

Find the equation of the line of the below a) gradient 2 passes through (0,8) b) gradient 4 passes through (0,3) 5. Divide a) (2x³4x² + 6x - 3) ÷ (x+4) b) (6x³ - 4x² + 3x + 2) ÷ (2x + 2)

Step-by-step solution

Identify the gradient and the y-intercept.

Substitute the values into the equation $y = mx + c$ .

Identify the gradient and the y-intercept.

Substitute the values into the equation $y = mx + c$ .

Divide $2x^3$ by $x$ .

Divide $-4x^2$ by $x$ .

Divide $22x$ by $x$ .

Divide $6x^3$ by $2x$ .

Divide $-10x^2$ by $2x$ .

Divide $13x$ by $2x$.

Still stuck on this one?

More Mathematics Questions

Find the equation of the line of the below a) gradient 2 passes through (0,8) b) gradient 4 passes through (0,3) 5. Divide a) (2x³4x² + 6x - 3) ÷ (x+4) b) (6x³ - 4x² + 3x + 2) ÷ (2x + 2)

Step-by-step solution

Identify the gradient and the y-intercept.

Substitute the values into the equation y=mx+cy = mx + cy=mx+c.

Identify the gradient and the y-intercept.

Substitute the values into the equation y=mx+cy = mx + cy=mx+c.

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Divide $13x$ by $2x$.

Still stuck on this one?

More Mathematics Questions

Substitute the values into the equation $y = mx + c$ .

Substitute the values into the equation $y = mx + c$ .

Divide $2x^3$ by $x$ .

Divide $-4x^2$ by $x$ .

Divide $22x$ by $x$ .

Divide $6x^3$ by $2x$ .

Divide $-10x^2$ by $2x$ .