Consider the following expanded powers of (a + b)n, where a + b is any binomial and n is a whole number. Look for patterns.
Each expansion is a polynomial. There are some patterns to be noted.
1. There is one more term than the power of the exponent, n. That is, there are terms in the expansion of (a + b)n.
2. In each term, the sum of the exponents is n, the power to which the binomial is raised.
3. The exponents of a start with n, the power of the binomial, and decrease to 0. The last term has no factor of a. The first term has no factor of b, so powers of b start with 0 and increase to n.
4. The coefficients start at 1 and increase through certain values about "half"-way and then decrease through these same values back to 1.
The fundamental theorem of algebra states that a polynomial with degree n has at most n solutions. The "at most" depends on the fact that the solutions might not all be real number.
In fact, if you use complex number, then a polynomial with degree n has exactly n roots.
So, in particular, a third-degree polynomial can have at most 3 roots.
In fact, in general, if the polynomial 
has solutions 
, then you can factor it as
So, a third-degree polynomial can't have 4 (or more) solutions, because otherwise you could write it as
But this is a fourth-degree polynomial.
Q1. The answers are (–1, 8), (0, 7), (3, 18)
<span>–3x + y ≥ 7
</span>Let's go through all choices:
<span>(–2, –3)
</span>(-3) * (-2) + (-3) ≥ 7
6 - 3 ≥ 7
3 ≥ 7 INCORRECT
(–1, 8)
(-3) * (-1) + 8 ≥ 7
3 + 8 ≥ 7
11 ≥ 7 CORRECT
(0, 7)
(-3) * 0 + 7 ≥ 7
0 + 7 ≥ 7
7 ≥ 7 CORRECT
(1, 9)
(-3) * 1 + 9 ≥ 7
-3 + 9 ≥ 7
6 ≥ 7 INCORRECT
(3, 18)
(-3) * 3 + 18 ≥ 7
-9 + 18 ≥ 7
9 ≥ 7 CORRECT
Q2. The answers are:
5x + 12y ≤ 80
x ≥ 4
<span>y ≥ 0
</span>
<span>x - small boxes
</span><span>y - large boxes
</span>He has x small boxes that weigh 5 lb each and y large boxes that weigh 12 lb each <span>on a shelf that holds up to 80 lb:
5x + 12y </span>≤ 80
Jude needs at least 4 small boxes on the shelf: x ≥ 4
Let's check if y can be 0:
5x + 12y ≤ 80
5x + 12 * 0 ≤ 80
5x + 0 ≤ 80
5x ≤ 80
x ≤ 80 / 5
x ≤ 16
x ≥ 4 can include x ≤ 16
So, y can be 0: y ≥ 0
I worked it out and I got south Africa
A "perfect square" is an integer that is the square of an integer.
The largest 6-digit integer is 999,999.
The square root of it is 999.9995
So the greatest square of an integer is the square of 999 = <u>998,001</u> .
