Unknown

118 3. Factors and Zeros

6.8. For example

1 1

-=t-1+

Bt3+Ct2+Dt+E

t5 - 1 @+P+P+t+l

for suitable A, B, C, D, E.

7.1. (c) Rearrange terms; collect terms with factor x + y.

(d) Set z = 1 and compare with (c). Alternatively, set ty = 1.

(e) Set b = c.

(g) Set a + b = 0.

(i) Set z equal to an imaginary cube root of unity.

(j) Try a substitution z = ax.

(k) Express as a difference of squares.

(m) Set c = 0, a + b + c = 0. Four linear factors are easily found.

(p) Factor the polynomial determined by setting t = 0. Alternatively,

express in terms of the elementary symmetric functions x + y + z,

xy + yr + 22, xyz (see Exercise 2.2.13).

(q) The polynomial vanishes when (x,y,z, w) = (l,l, 1,l). What

possible linear factors does this suggest?

(s) Write as a polynomial in z 2, Is there a substitution for z2 as a

function of x and y which will make the polynomial vanish?

(u) Factor as a difference of squares.

7.3. What should p(l), p(2), p(3) be?

7.4. The polynomial should vanish when z = i.

7.5. (a) The zeros of the quadratic are reciprocals; the sum of the zeros

of the cubic is 0. Use Exercise 1.5.6.

(b) The zeros of cx3 + bx2 + a are the reciprocals of the zeros of

ax3 + bx + c.

7.6. Use the method of undetermined coefficients. It can be arranged that

the constant term in each linear factor is 1.

7.8. x2 - x + a must divide xl3 + 2 + 90 when x = 0,l. The case a = -2

can be eliminated almost at once; factor the quadratic.

7.9. To get an idea of what the cube root might be, look at the situation

in which any one of the variables is set equal to 0. Make a conjecture

and check it out using the Factor Theorem. Can you manipulate the

polynomial directly to reveal that it is a cube?