worksheet 1.4

1. for $a{x}^2+bx+c=10x^2-7$, find the values of a, b and c
   $a=10$
   $b=0$
   $c=-7$

2. for $(2a-b)x^2+(a+2b)x+8=4x^2-3x+8$, find values of a and b
   $2a-b=4$
   $a+2b=-3$
   $a=-3-2b$
   $-6-5b=4$
   $b=-2$
   $a=1$

3. for $(2a-3b)x^2+(3a+b)x+c=7x^2+5x+7$, find the values of a, b and c
   $2a-3b=7$
   $3a+b=5$
   $c=7$
   $b=5-3a$
   $2a-15+9a=7$

4. for $2x^2+4x+5=a(x+b{)}^2+c$, find the values of a, b and c
   $2x^2+4x+5=a(x^2+2bx+b^2)+c$
   $2x^2+4x+5=a{x}^2+2abx+a{b}^2+c$
   $a=2$
   $2ab=4$
   $a{b}^2+c=5$
   $b=1$
   $c=3$

5. express $x^2$ in the form $c(x+2{)}^2+a(x+2)+d$
   $x^2=c(x^2+4x+4)+ax+2a+d$
   $x^2=c{x}^2+4cx+4c+ax+2a+d$
   $x^2=c{x}^2+(4c+a)x+4c+2a+d$
   $c=1$
   $4c+a=0$
   $4c+2a+d=0$
   $a=-4$
   $d=4$
   so the answer is $1(x+2{)}^2+-4(x+2)+4$

6. express $x^3$ in the form $(x+1{)}^3+a(x+1{)}^2+b(x+1)+c$
   $x^3=x^3+3x^2+3x+1+a(x^2+2x+1)+bx+b+c$
   $x^3=x^3+$

7. find the values of a, b and c such that $x^2=a(x+1{)}^2+bx+c$
   $=a(x^2+2x+1)+bx+c$
   $=a{x}^2+2ax+a+bx+c$
   $=a{x}^2+(2a+b)x+a+c$
   a=1
   2a+b=0
   b=-2
   a+c=0
   c=-1

8. 1. show that $3x^3-9x^2+8x+2$ cannot be expressed in the form $a(x+b{)}^3+c$
      $a(x+b{)}^3+c=a(x^3+3x^{2}b+3x{b}^2+b^3)+c$
      $=a{x}^3+3a{x}^{2}b+3ax{b}^2+a{b}^3+c$
      $a=3$
      $a{b}^3+c=2$
      $3a{b}^2=8$
      $3ab=-9$
      $b=-1$
      See that $3a{b}^2=8$ is false. So cannot.
   2. if $3x^2-9x^2+9x+2$ can be expressed in the form $a(x+b{)}^3+c$, then find the values of a, b and c
      $a(x+b{)}^3+c=a(x^3+3x^{2}b+3x{b}^3+b^3)+c$
      $=a{x}^3+3a{x}^{2}b+3ax{b}^2+a{b}^3+c$
      $a=3$
      $3ab=-9$
      $3a{b}^2=9$
      $a{b}^3+c=2$
      $b=-1$
      $-3+c=2$
      $c=5$

9. show that constants a, b, c, and d can be found such that $n^3=a(n+1)(n+2)(n+3)+b(n+1)(n+2)+c(n+1)+d$
   $n^3=a(n+1)(n^2+5n+6)+bn(n^2+3n+2)+cn+c+d$
   $n^3=a+a{n}^3+a5n^2+a6n+a{n}^2+a5n+a6+b{n}^3+b3n^2+b2n+cn+c+d$
   $n^3=a{n}^3+(6a+b)n^2+(11a+3b+c)n+6a+2b+c+d$
   $a=1$
   $6a+b=0$
   $b=-6$
   $11a+3b+c=0$
   $c=7$
   $6a+2b+c+d=0$
   $d=-1$

10. 1. show that no constants a and b can be found such that $n^2=a(n+1)(n+2)+b(n+1)(n+2)+c(n+1)+d$
    2. express $n^2$ in the form $a(n+1)(n+2)+b(n+1)+c$
11. 1. express $a(x+b{)}^2+c$ in expanded form
    2. express $a{x}^2+bx+c$ in completed square form

12. prove that, if $a{x}^3+b{x}^2+cx+d=(x+1{)}^2(px+q)$, then $b=d-2a$ and $c=a-2d$

13. if $3x^2+10x+3=c(x-a)(x-b)$ for all values of $x$, find the values of a, b and c

14. for any number $n$, show that $n^2$ can be expressed as $a(n-1{)}^2+b(n-2{)}^2+c(n-3{)}^2$, and find the values of a, b and c

15. if $x^3+2x^2-9x+c$ can be expressed in the form $(x-a{)}^2(x-b)$, show that either $c=5$ or $c=-27$, and find a and b for each of these cases.

16. A polynomial $P$ is said to be even if $P(-x)=P(x)$ for all x. A polynomial $P$ is said to be odd if $P(-x)=-P(x)$ for all x.

    1. show that, if $P(x)=a{x}^4+b{x}^3+c{x}^2+dx+e$ is even, then $b=d=0$
    2. show that, if $P(x)=a{x}^5+b{x}^4+c{x}^3+d{x}^2+ex+f$ is odd, then $b=d=f=0$