Graph of Polynomial

A Polynomial function is an algebraic expressions that satisfied quadratic function, cubic function format ...etc

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How to solve Polynomial using the graph

The diagram below shows the graph of $Y\; =x^3+x^2-\mathrm{5x}+3$

Use the graph

(i) to calculate an estimate of the gradient of the curve at the point (2,5)

(ii) to solve the equations

(a) $x^3+x^2-\mathrm{5x}+3=\; 0$

(b) $x^3+x^2-\mathrm{5x}+3=\; 5x$

(iii) to calculate an estimate of the area bounded by the curve, x = 0, y = 0 and x = -2

Solutions

(i)

Use the graph to find gradient

(2, 5) (2.5, 10)

Gradient = $\frac{\mathrm{Y2\; -\; Y1}}{\mathrm{X2\; -\; X1}}$

X1 = 2, X2 = 2.5, Y1 = 5, Y2 = 10

Gradient = $\frac{\mathrm{10\; -\; 5}}{\mathrm{2.5\; -\; 2}}$

Gradient = $\frac{5}{0.5}$

Answer: Gradient = $10$

(ii)

(a) $x^3+x^2-\mathrm{5x}+3=\; 0$

From the equation above $Y\; =x^3+x^2-\mathrm{5x}+3$.

Therefore $y=\; 0$

Draw a line $y=\; 0$ on the graph. hence find the values of x at the points the line intersect the graph

values of x at the points the line $y=\; 0$ touchs the graph

Answer: (1, 0) and (-3, 0) from line $y=\; 0$

(b) $x^3+x^2-\mathrm{5x}+3=\; 5x$

$y\; =x^3+x^2-\mathrm{5x}+3$

$y=\; 5x$

Plot the line $y=\; 5x$ on the graph

given that x = -3 , y = 5x (-3, -15) and x = 3, y = 5x (3, 15)

values of x at the points the line $y=\; 5x$ touchs the graph

Answer: (-4, -20) and (0.2, 0.2) and (2.5, 16)

(iii)

Area bounded by lines x = 0, y = 0 and x = -2

The shape of the area bounded by lines x = 0, y = 0 and x = -2 is a tripezium

Formula for area of a tripezium

Area = $\frac{\mathrm{a\; +\; b}}{2}$ h

a = 2, b = 1, h = 9

Area = $\frac{\mathrm{2\; +\; 1}}{2}$ 9

Area = $\frac{3}{2}$ 9

Area = $\frac{\mathrm{27}}{2}$

Answer: Area = $13.5{\mathrm{units}}^2$

How to solve Polynomial