IBHM 086 107.pdf


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4 Polynomials

Example
Factorise fully g1x2 ⫽ 2x4 ⫹ x3 ⫺ 38x2 ⫺ 79x ⫺ 30.
Without a calculator, we need to guess a possible factor of this polynomial.
Since the constant term is ⫺30, we know that possible roots are
;1, ;2, ;3, ;5, ;6, ;10, ;15, ;30.
We may need to try some of these before finding a root. Normally we would
begin by trying the smaller numbers.
1

2

1

⫺38

⫺79

⫺30

2


2
3


3
⫺35


⫺35
⫺114


⫺114
⫺144

Clearly x ⫺ 1 is not a factor.
Trying x ⫽ ⫺1 and x ⫽ 2 also does not produce a value of 0. So we need to
try another possible factor. Try x ⫹ 2.
⫺2

2

2

1

⫺38

⫺79

⫺30


⫺4


6


64


30

⫺3

⫺32

⫺15

0

So x ⫹ 2 is a factor.
Now we need to factorise 2x3 ⫺ 3x2 ⫺ 32x ⫺ 15. We know that x ⫽ ;1
do not produce factors so we try x ⫽ ⫺3.
⫺3

2

2

⫺3

⫺32

⫺15


⫺6


27


15

⫺9

⫺5

0

Hence g1x2 ⫽ 1x ⫹ 32 1x ⫹ 22 12x2 ⫺ 9x ⫺ 52
⫽ 1x ⫹ 32 1x ⫹ 22 12x ⫹ 12 1x ⫺ 52

Exercise 2
1 Show that x ⫺ 3 is a factor of x2 ⫹ x ⫺ 12.
2 Show that x ⫺ 3 is a factor of x3 ⫹ 2x2 ⫺ 14x ⫺ 3.
3 Show that x ⫺ 2 is a factor of x3 ⫺ 3x2 ⫺ 10x ⫹ 24.
4 Show that 2x ⫺ 1 is a factor of 2x3 ⫹ 13x2 ⫹ 17x ⫺ 12.
5 Show that 3x ⫹ 2 is a factor of 3x3 ⫺ x2 ⫺ 20x ⫺ 12.
6 Show that x ⫹ 5 is a factor of x4 ⫹ 8x3 ⫹ 17x2 ⫹ 16x ⫹ 30.

92

We do not need to use
division methods to
factorise a quadratic.