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Polynomials, EXT1 EQ-Bank 11

The polynomial  \(R(x)=x^3+p x^2+q x+6\)  has a double zero at  \(x=-1\)  and a zero at  \(x=s\).

Find the values of \(p, q\) and \(s\).   (3 marks)

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\(s=-6, \ p=8, \ q=13\)

Show Worked Solution

\(R(x)=x^3+p x^2+q x+6\)

\(R(x)\ \text{is monic with a zero at} \ s \ \text{and double zero at}\ -1:\)

\(R(x)\) \(=(x+1)^2(x-s)\)
  \(=\left(x^2+2 x+1\right)(x-s)\)
  \(=x^3+2 x^2+x-s x^2-2 s x-s\)
  \(=x^3+(2-s) x^2+(1-2 s) x-s\)

 

\(\text{Equating coefficients:}\)

\(-s=6 \ \Rightarrow \ s=-6\)

\(p=2-(-6)=8\)

\(q=1-2(-6)=13\)

Polynomials, EXT1 EQ-Bank 10

The polynomial  \(R(x)=2 x^4+a x^3+b x^2+c x+d\)  has a double zero at  \(x=1\), a zero at  \(x=-3\), and passes through the point \((0,-12)\).

Find the integer values of \(a, b, c, d\) and the fourth zero of the polynomial.   (4 marks)

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\(a=-2, \ b=-14, \ c=26, \ d=-12\)

\(\text{Fourth zero:} \ \ x=2\)

Show Worked Solution

\(R(x)=2 x^4+a x^3+b x^2+c x+d\)

\(\text{Since leading coefficient is 2 with a double zero at 1 and a zero at }-3:\)

\(R(x)=2(x-1)^2(x+3)(x-k) \ \ \text{where} \ k \ \text{is the fourth zero.}\)

\(\text{The polynomial passes through}\ (0,-12):\)

\(R(0)=2(0-1)^2(0+3)(0-k)=-12\ \ \Rightarrow\ \ k=2\)
 

\(\text{Expanding}\ R(x):\)

\(R(x)\) \(=2(x-1)^2(x+3)(x-2)\)  
  \(=2\left(x^2-2 x+1\right)(x+3)(x-2) \)  
  \(=2(x^3+3 x^2-2 x^2-6 x+x+3)(x-2) \)  
  \(=2(x^3+x^2-5 x+3)(x-2) \)  
  \(=2(x^4+x^3-5 x^2+3 x-2 x^3-2 x^2+10 x-6) \)  
  \(=2 x^4-2 x^3-14 x^2+26 x-12\)  

 

\(\text{Equating coefficients:}\)

\(a=-2, \ b=-14, \ c=26, \ d=-12\)

\(\text{Fourth zero:} \ \ x=2\)

Polynomials, EXT1 EQ-Bank 3

Consider the polynomial \(P(x)=x(3-x)^3\).

  1. State the degree of the polynomial and identify the leading coefficient.   (1 mark)

  2. Explain what happens to \(y\) as  \(x \rightarrow \pm \infty\).   (1 mark)

  3. Without using calculus, sketch \(P(x)\) showing its general form and any \(x\)-intercepts.   (2 marks)

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a.    \(\text{Degree}\ P(x)=4\)

\(\text{Leading co-efficient}=-1\)
 

b.    \(\text{As} \ \ x \rightarrow-\infty,-x^4 \rightarrow-\infty, \ y \rightarrow-\infty\).

\(\text{As} \ \ x \rightarrow \infty,-x^4 \rightarrow-\infty, \ y \rightarrow-\infty\).
 

c.    \(P(x)\ \text{has zeroes at}\ \ x=0, 3:\)


       

Show Worked Solution

a.    \(\text{Degree}\ P(x)=4\)

\(\text{Leading co-efficient}=-1\)
 

b.    \(\text{As} \ \ x \rightarrow-\infty,-x^4 \rightarrow-\infty, \ y \rightarrow-\infty\).

\(\text{As} \ \ x \rightarrow \infty,-x^4 \rightarrow-\infty, \ y \rightarrow-\infty\).
 

c.    \(P(x)\ \text{has zeroes at}\ \ x=0, 3:\)


       

Polynomials, EXT1 EQ-Bank 4

The polynomial  \(p(x) = x^3 + ax^2 + b\)  has a zero at \(r\) and a double zero at 4.

Find the values of \(a, b\) and \(r\).   (3 marks)

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\(a =-6, b = 32, r = -2\)

Show Worked Solution

\(p(x) = x^3 + ax^2 + b\)

\(\text{Zero at \(r\) and double zero at 4:}\)

\(p(x)\) \(=(x-4)^2(x-r) \)  
  \(=(x^2-8x+16)(x-r)\)  
  \(=x^3-8x^2+16x-rx^2+8rx-16r\)  
  \(=x^3+(-8-r)x^2+(16+8r)x-16r\)  

 

\(\text{Equating coefficients:}\)

\(16+8r=0\ \ \Rightarrow \ \ r=-2\)

\(a=-8-(-2)=-6\)

\(b=-16 \times -2=32\)