Polynomials Test - 3

1. p(x) is the remainder obtained when polynomial p(y) is divided by y – x.

Explanation: Based on remainder theorem, here p(y) is an unnamed polynomial and y - x is a linear factor.
y - x = 0
So, y = x
p(y) = p(x)
So, the remainder obtained is p(x).

2. y – b is a factor of polynomial p(y), if b is a zero of p(y).

Explanation: Using the factor theorem, p(y) is a polynomial of degree n > 1, b is any real number and p(b) = 0, then y - b is a factor of p(y).

3. Every real number is a zero of the polynomial.

Explanation: Every polynomial has distinct roots. Every real number cannot be a root of every polynomial.

4. A term with no letter in polynomial is called a constant.

Explanation: It is because it does not contain any modifiable variables in which values can be substituted to get a new answer.

Let p(z) = -2z⁴ - 7z³ - 3z² - az - 10
Since (z + 2) is a factor of the polynomial p(z);

p(-2) = 0
-2(-2)⁴ - 7(-2)³ - 3(-2)² - a(-2) - 10 = 0

-2(16) - 7(-8) - 3(4) - a(-2) - 10 = 0
-32 + 56 - 12 + 2a - 10 = 0

2a + 2 = 0
a = -1

a⁸ - b⁸
= (a⁴)² - (b⁴)²

= (a⁴ - b⁴)(a⁴ + b⁴)
= [(a²)² - (b²)²)](a⁴ + b⁴)

= (a² - b²)(a² + b²)(a⁴ + b⁴)
= (a + b)(a - b)(a² + b²)(a⁴ + b⁴)

Let f(a) = a³ - b³

We know that when f(a) is divided by a - b, then by remainder theorem, the remainder will be f(b).

So, f(b) = b³ - b³ = 0

We have p(y) = y³ + ay² - 2y + a

From Remainder Theorem, we know that when p(y) is divided by (y + a), the remainder is p(-a).

Now, p(-a) = (-a)³ + a(-a)² - 2(-a) + a = -a³ + a³ + 2a + a = 3a