EXERCISES AND PROBLEMS

16 Break these polynomials down into factors: a) x 2 + 4x – 5 b) x 2 + 8x + 15 c) 7x 2 – 21x – 280 d) 3x 2 + 9x – 210 e) 2x 2 – 9x – 5 f ) 3x 2 – 2x – 5 g) 4x 2 + 17x + 15 h) –x 2 + 17x – 72

17 Factorise the following polynomials and give their roots: a) x 3 + 2x 2 – x – 2 b) 3x 3 – 15x 2 + 12x c) x 3 – 9x 2 + 15x – 7 d) x 4 – 13x 2 + 36

Algebraic fractions

18 Simplify these algebraic fractions

Practice Makes Perfect

25 Divide and express these divisions like this: divisor dividend = quotient + divisor remainder a) (3x5 – 2x 3 + 4x – 1) : (x 3 – 2x + 1) b) (x 4 – 5x 3 + 3x – 2) : (x 2 + 1) c) (4x5 + 3x 3 – 2x) : (x 2 – x + 1) d) (x 3 – 5x 2 + 3x + 1) : (x 2 – 5x + 1)

26 Perform the following divisions: a) (2x 3 – x 2 + 3x – 1) : (2x 2 + 2x) b) (6x 4 – x 3 – 3x + 1) : (2x 2 – 1) c) (x 5 – 3x 2 – 2x – 5) : (x 3 + 4x – 1)

19 Check that these pairs of fractions are equivalent: a) x x 312 4 3 1 ––y b) x xx x 22 y 2 + c) xy xy xy 1 –y – 22 + d) xx x x 22 2 –y – 2

20 Write an equivalent fraction in each case: a) x 1 –3 + b) x 5 21 + c) x x 2 1 –+

21 Find P (x) so that the fractions are equivalent

22 Break down into factors and simplify.

() x x 3 9 –2 2 + b) x x 4 2 –2 + x xx 25 25 10 ––2 2 + d) xxyy xxy 2 –

–22 2 + e) xx x 6 2 ––2 + f ) xy xy xy 2 3 –2 22 a) x x x 2 1 4 11 –+ b) x xx 1 3 11 –2 + c) x x 2 3 1 – + d) x x x 2 3 1 –2 + e) x x x 3 3 –– f ) x x x x 1 3 3 ––+ +

23 Reduce to a common denominator and calculate.

24 Calculate and simplify where possible: a) x xx 4 3 2 –· 2 b) xx 8 21 6 : – c) xx x x 3 9 2 · ––22 d) : x x x x 1 1 1 3 –2 + +

27 Apply Ruffini’s rule to find the quotient and remainder of the following divisions: a) (4x 2 – 8x + 3) : (4x – 2) b) (2x 3 – 4x 2 + 3x – 2) : (2x – 3) c) (3x 3 – 2x – 1) : (3x + 1)

28 Break down the following polynomials into factors: a) (x 2 – 25)(x 2 – 6x + 9) b) (x 2 – 7x)(x 2 – 13x + 40)

29 Factorise the following polynomials and give their roots: a) x 3 – 2x 2 – 2x – 3 b) 2x 3 – 7x 2 – 19x + 60 c) x 3 – x – 6 d) 4x 4 + 4x 3 – 3x 2 – 4x – 1 e) 6x 3 + 13x 2 – 4 f ) 4x 3 + 12x 2 – 25x – 75

30 Break down the following polynomials into factors and give their roots: a) x 4 – 2x 2 + 1 b) x 3 – 2x 2 – 9x + 18 c) x 4 – x 3 – 7x 2 + x + 6 d) 8x 3 + 6x 2 – 11x – 3 e) 3x 3 + 8x 2 + 3x – 2 f ) x 3 – 2x 2 + 2x – 4 a) () x xm x 15 1 –:)( 2 ++ b) () ): ( xx x mx 2 2 –2 3 ++ + c) ) xxmx mx62 2 – () :( 2 3 + c) xx x 4 21 12 1 –––2 cm d) : x y y x 11++c c m m

31 Find the value of m that makes each of the following divisions exact.

32 Find m so that () ( Px xx mx m 52 ) =+32 ++ is divisible by x – 1.

33 Break down the dividend and the divisor into factors, then simplify.

40 Replace the dots with expressions that make the fractions equivalent.

SOLVE SIMPLE PROBLEMS

UNDERSTAND AND APPLY IN THE CHALLENGE

39 Calculate a, b and c to verify the

41 Express the following polynomials in the form ax 3 + bx 2 + cx + d: a) () ) x xx22 25 – )( ( 2 3 ++ + b) () ) )( ( xx x 3 32 3 –32 c) )) x x 14 13 2 – (( 3 +

42 Write a third-degree polynomial that has the given roots in each case: a) 0, 1 and 2 b) –1 and 3 c) 0 and 5

43 For each of the following, write a polynomial that meets the given condition: a) Fourth-degree without roots. b) Has two double roots, 2 and –2. c) Third-degree with a single root. d) Fourth-degree and with three roots.

44 Calculate the value of the dividend in each of the following cases: a) Divisor: (x 2 – 2). Quotient: (x + 3). Remainder: 7 b) Divisor: (x 2 + 2x). Quotient: (x 2 + x – 2). Remainder: (–3x + 4) c) Divisor: (x 2 + 1). Quotient: (x – 2). Remainder: (2x + 10)

45 Calculate the value of m that makes the polynomial mx 3 – 3x 2 + 5x + 9m divisible by x + 2.

46 Calculate the value of m and n that makes the polynomial P (x) = x 3 – m x 2 + n x + 4 divisible by x – 2 and x + 2. What are the roots of P (x)?

47 Calculate the values a and b that make the polynomial P (x) = x 4 – 2x 3 + ax 2 + bx + 15 divisible by (x + 3) and by (x – 5).

48 Calculate the value of m that gives the following divisions the remainder indicated: a) (x 3 – 2x 2 – x + m) : (x + 1) Remainder = –1 b) (2x 3 – 12x + 2m) : (x – 3) Remainder = –5

49 The remainder of this division is – 8: (2x 4 + kx 3 – 7x + 6) : (x – 2) What is the value of k?