Worksheet 4.7 Polynomials
Worksheet 4.7 Polynomials
Worksheet 4.7 Polynomials
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Solution a) By trial and error notice that<br />
p(2) = 48 − 66 + 22 − 2 = 0<br />
i.e. 2 is a root of p(x).<br />
So x − 2 is a factor of p(x).<br />
To find other factors we’ll divide p(x) by (x − 2).<br />
6x 2 − 5x + 1<br />
x − 2 6x 3 − 17x 2 + 11x − 2<br />
6x 3 − 12x 2<br />
−5x 2 + 11x<br />
−5x 2 + 10x<br />
x − 2<br />
x − 2<br />
0<br />
So p(x) = (x − 2)(6x 2 − 5x + 1). Now notice<br />
6x 2 − 5x + 1 = 6x 2 − 3x − 2x + 1<br />
= 3x(2x − 1) − (2x − 1)<br />
= (3x − 1)(2x − 1)<br />
So p(x) = (x − 2)(3x − 1)(2x − 1) and its factors are (x − 2), (3x − 1) and (2x − 1).<br />
Solution b) The solutions to p(x) = 0 occur when<br />
That is,<br />
Exercises:<br />
x − 2 = 0, 3x − 1 = 0, 2x − 1 = 0.<br />
x = 2, x = 1<br />
1<br />
, x =<br />
3 2 .<br />
1. For each of the following polynomials find (i) its factors; (ii) its roots.<br />
(a) x 3 − 3x 2 + 5x − 6<br />
(b) x 3 + 3x 2 − 9x + 5<br />
(c) 6x 3 − x 2 − 2x<br />
(d) 4x 3 − 7x 2 − 14x − 3<br />
2. Given that x − 2 is a factor of the polynomial x 3 − kx 2 − 24x + 28, find k and the roots<br />
of this polynomial.<br />
6
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