Given two points (3, 7) and (6, 8) with a line drawn through the points, calculate the slope of the line

Mathematics

2 answers:

irinina [24]3 years ago

5 0

Answer:

$= \frac{1}{3}$

Step-by-step explanation:

$(3 \: \: \: \: \: 7) = > (x1 \: \: \: \: \: y1) \\ (6 \: \: \: \: \: 8) = > (x2 \: \: \: \: \: y2)$

$slope \\ = \frac{y1 - y2}{x1 - x2} \\ = \frac{7 - 8}{3 - 6} \\ = \frac{ - 1}{ - 3} \\ = \frac{1}{3}$

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vredina [299]3 years ago

3 0

Answer:

1/3

Step-by-step explanation:

Given two points on the line, we can find the slope using

m = (y2-y1)/(x2-x1)

= (8-7)/(6-3)

= 1/3

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Use the Factor Theorem and synthetic division to show x + 5 is a factor of f(x) = 2x3 + 7x2 − 14x + 5

olga2289 [7]

Answer:

Factor theorem: $f(-5) = 0$ .

Synthetic division: $f(x) = (x + 5)\, (2\, x^2 -3\, x + 1)$ .

Step-by-step explanation:

<h3>Factor Theorem</h3>

By the factor theorem, a monomial of the form $(x - a)$ (where $a$ is a constant) is a factor of polynomial $f(x)$ if and only if $f(a) = 0$ .

In this question, the monomial is $(x + 5)$ , which is equivalently $(x - (-5))$ . $a = -5$ .

$\begin{aligned}& f(-5) \\ &= 2 \times (-5)^{3} + 7 \times (-5)^{2}- 14\times (-5) + 5\\ &= -250 + 175 + 70 + 5 \\ &= 0\end{aligned}$ .

Hence, by the factor theorem, $(x + 5)$ , which is equivalent to $(x - (-5))$ , is a factor of $f(x)$ .

<h3>Synthetic Division</h3>

$\begin{aligned}& f(x) \\ &= 2\, x^{3} + 7\, x^{2} - 14\, x + 5 \\ &= \underbrace{(x + 5) \, (2\, x^2)}_{2\, x^{3} + 10\, x^{2}} - 3\, x^{2} - 14\, x + 5 \\ &= \underbrace{(x + 5) \, (2\, x^2)}_{2\, x^{3} + 10\, x^{2}} + \underbrace{(x + 5)\, (-3\, x)}_{-3\, x^{2} - 15\, x} + (x + 5) \\ &= (x + 5)\, (2\, x^{2} - 3\, x + 1)\end{aligned}$ .