p(x)=8x^(2)-19x-15

Question

Find the zeroes of the following polynomials and verify the relationship between the zeroes and the coefficient of the polynomial: P(x)=8x^(2)-19x-15

Answer 1

f(x)=8x²-19x-15.

Let f(x)=(ax-b)(cx+d), so we have to find a, b, c, d.

Here's how we do it.

Factors of 8=(a,c)={(1,8), (2,4), (4,2), (8,1)}; factors of 15=(b,d)={(1,15), (3,5), (5,3), (1,15)}.

We have (ax-b)(cx+d)=acx²+x(ad-bc)-bd, where ac=8 and bd=15. We want ad-bc=-19.

The table below shows all possibilities. The red row gives us the answer.

f(x)=(x-3)(8x+5), zeroes are x=-⅝ and 3, because either 8x+5=0, 8x=-5, x=-⅝; or x-3=0, x=3.

You can see that this red row is similar to the second to last row in the table but the + and - signs are reversed. We could have used a table half the size and then sorted out the signs (which would always be different) because the sign of the constant term, -15, is negative.

a	b	c	d	ad	bc	ad-bc
1	1	8	15	15	8	7
1	3	8	5	5	24	-19
1	5	8	3	3	40	-37
1	15	8	1	1	120	-119
2	1	4	15	30	4	26
2	3	4	5	10	12	-2
2	5	4	3	6	20	-14
2	15	4	1	2	60	-58
4	1	2	15	60	2	58
4	3	2	5	20	6	14
4	5	2	3	12	10	2
4	15	2	1	4	30	-26
8	1	1	15	120	1	119
8	3	1	5	40	3	37
8	5	1	3	24	5	19
8	15	1	1	8	15	-7