+0

# The polynomial has degree 3. If , , , and , then what are the -intercepts of the graph of ?

0
207
1

The polynomial  has degree 3. If , and , then what are the -intercepts of the graph of ?

Please explain very well in this question. I am so sorry! I just need a little help.

Guest Dec 4, 2014

#1
+18827
+10

----------------------------------------------------------

The polynomial  has degree 3. If , and , then what are the x-intercepts of the graph of ?

$$\small{\text{The polynomial f(x) of degree 3 is }} f(x) = ax^3+bx^2+cx+d$$

I.  We need a, b, c and d :

$$\small{\text{ \begin{array}{r|r|lrclrclccl} \hline x & y & &f(x)& =& ax^3+bx^2+cx+d & && &{d}&{=}&{0} \\ \hline -1 & 15& (1) & 15 &=& a(-1)^3+b(-1)^2+c(-1) +d & 15&=& -a+b-c+d & 15&=&-a+b-c\\ 0 & 0 & (2) & 0 &=& a(0)^3+b(0)^2+c(0) +d & {0} &{=}& {d} & -5&=&a+b+c\\ 1 & -5 & (3) & -5 &=& a(1)^3+b(1)^2+c(1) +d & -5 &=& a+b+c+d & \\ 2 & 12 & (4) & 12 &=& a(2)^3+b(2)^2+c(2) +d & 12 &=& 8a+4b+2c+d & 12 &=& 8a+4b+2c\\ \hline \end{array} }}$$

d=0:

(1) -a + b - c = 15

(2)  a + b + c = -5

(4) 8a+4b+2c = 12 | :2    $$\Rightarrow$$   (4) 4a + 2b + c = 6

----------------------------------------------------------

(1)+(2): 2b = 10  $$\Rightarrow$$  $${ b = 5}$$

----------------------------------------------------------

b=5:

(1)   a + c = -10

(2)   a + c = -10

(4) 4a + c =  -4

----------------------------------------------------------

(4)-(2): 3a = -4 -(-10) = 6  $$\Rightarrow$$  3a = -4+10  $$\Rightarrow$$  3a = 6   =>  $${a=2}$$

(1)  2 + c = -10  $$\Rightarrow$$  $${c = -12}$$

$$\small{\text{The polynomial f(x) of degree 3 is }} f(x) = 2x^3+5x^2-12x+0$$

II.  x-intercepts of the graph of $$f(x)$$?

$$2x^3+5x^2-12x = 0 \\ \underbrace{x}_{=0}*\underbrace{( 2x^2+5x-12 )}_{=0} = 0 \\\\ {x_1 = 0} \\\\ 2x^2+5x-12 = 0 \quad | \quad{ ax^2+bx+c=0 => x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} } \\\\ x_{2,3}=\frac{-5\pm\sqrt{25-4*2*(-12)} }{4} \\\\ x_{2,3}=\frac{-5\pm\sqrt{121} }{4} \\\\ x_{2,3}=\frac{-5\pm\11 }{4} \\\\ x_2=\frac{-5+11 }{4} = \frac{6}{4} = 1.5 \quad \Rightarrow \quad {x_2=1.5} \\\\ x_3=\frac{-5-11 }{4} = \frac{-16}{4} = -4 \quad \Rightarrow \quad {x_3=-4} \\\\$$

The x-intercepts are: -4, 0 and 1.5

heureka  Dec 4, 2014
Sort:

#1
+18827
+10

----------------------------------------------------------

The polynomial  has degree 3. If , and , then what are the x-intercepts of the graph of ?

$$\small{\text{The polynomial f(x) of degree 3 is }} f(x) = ax^3+bx^2+cx+d$$

I.  We need a, b, c and d :

$$\small{\text{ \begin{array}{r|r|lrclrclccl} \hline x & y & &f(x)& =& ax^3+bx^2+cx+d & && &{d}&{=}&{0} \\ \hline -1 & 15& (1) & 15 &=& a(-1)^3+b(-1)^2+c(-1) +d & 15&=& -a+b-c+d & 15&=&-a+b-c\\ 0 & 0 & (2) & 0 &=& a(0)^3+b(0)^2+c(0) +d & {0} &{=}& {d} & -5&=&a+b+c\\ 1 & -5 & (3) & -5 &=& a(1)^3+b(1)^2+c(1) +d & -5 &=& a+b+c+d & \\ 2 & 12 & (4) & 12 &=& a(2)^3+b(2)^2+c(2) +d & 12 &=& 8a+4b+2c+d & 12 &=& 8a+4b+2c\\ \hline \end{array} }}$$

d=0:

(1) -a + b - c = 15

(2)  a + b + c = -5

(4) 8a+4b+2c = 12 | :2    $$\Rightarrow$$   (4) 4a + 2b + c = 6

----------------------------------------------------------

(1)+(2): 2b = 10  $$\Rightarrow$$  $${ b = 5}$$

----------------------------------------------------------

b=5:

(1)   a + c = -10

(2)   a + c = -10

(4) 4a + c =  -4

----------------------------------------------------------

(4)-(2): 3a = -4 -(-10) = 6  $$\Rightarrow$$  3a = -4+10  $$\Rightarrow$$  3a = 6   =>  $${a=2}$$

(1)  2 + c = -10  $$\Rightarrow$$  $${c = -12}$$

$$\small{\text{The polynomial f(x) of degree 3 is }} f(x) = 2x^3+5x^2-12x+0$$

II.  x-intercepts of the graph of $$f(x)$$?

$$2x^3+5x^2-12x = 0 \\ \underbrace{x}_{=0}*\underbrace{( 2x^2+5x-12 )}_{=0} = 0 \\\\ {x_1 = 0} \\\\ 2x^2+5x-12 = 0 \quad | \quad{ ax^2+bx+c=0 => x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} } \\\\ x_{2,3}=\frac{-5\pm\sqrt{25-4*2*(-12)} }{4} \\\\ x_{2,3}=\frac{-5\pm\sqrt{121} }{4} \\\\ x_{2,3}=\frac{-5\pm\11 }{4} \\\\ x_2=\frac{-5+11 }{4} = \frac{6}{4} = 1.5 \quad \Rightarrow \quad {x_2=1.5} \\\\ x_3=\frac{-5-11 }{4} = \frac{-16}{4} = -4 \quad \Rightarrow \quad {x_3=-4} \\\\$$

The x-intercepts are: -4, 0 and 1.5

heureka  Dec 4, 2014

### 6 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details