Polynomials (Chapter 6) 173Discussion
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Consider the general cubic p(x)=ax^3 +bx^2 +cx+d, a,b,c,d 2 R.What happens to p(x) ifxgets:
² very large and positive ² very large and negative?What does this tell you about the number of solutions that p(x)=0may have?Review set 6A
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1 Given p(x)=5x^2 ¡x+4 and q(x)=3x^2 +7x¡ 1 , find:
a p(x)+q(x) b 2 p(x)¡q(x) c p(x)q(x)2 Find the quotient and remainder of:a
2 x^2 +11x+18
x+3
b
x^3 ¡ 6 x^2 +10x¡ 9
x¡ 2
3 Find the zeros of:
a 3 x^2 +2x¡ 8 b x^2 +8x+114aGiven that x^3 +x^2 ¡ 3 x+9=(x+ 3)(ax^2 +bx+c), find the values ofa,b, andc.
b Show that x^3 +x^2 ¡ 3 x+9 has only one real zero.5 Use the Remainder theorem to find the remainder when:
a x^3 ¡ 4 x^2 +5x¡ 1 is divided by x¡ 2
b 2 x^3 +6x^2 ¡ 7 x+12 is divided by x+5.6 Use the Factor theorem to determine whether:
a x+1is a factor of 2 x^4 ¡ 9 x^2 ¡ 6 x¡ 1
b x¡ 3 is a factor of x^4 ¡ 2 x^3 ¡ 4 x^2 +5x¡ 6.7 2 x^2 +kx¡ 5 has remainder 3 when divided by x+4. Findk.8 ax^3 +5x^2 ¡x+b has remainder 7 when divided by x¡ 1 , and remainder¡ 11 when divided
by x+2. Findaandb.9 Findcgiven that x¡ 2 is a factor of x^5 ¡ 2 x^4 +cx^3 ¡ 7 x^2 +5x¡ 6.10 x¡ 4 is a factor of f(x)=x^3 +2x^2 +ax+b, and when f(x) is divided by x+2the
remainder is 18.
a Findaandb. b Find all zeros of f(x).11 Solve forx: x^3 ¡x^2 ¡ 17 x¡15 = 0Review set 6B
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a (3x^3 +2x¡5)(4x¡3) b (2x^2 ¡x+3)^24037 Cambridge
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Y:\HAESE\CAM4037\CamAdd_06\173CamAdd_06.cdr Thursday, 3 April 2014 5:19:18 PM BRIAN