Test: Polynomials in One Variable - ACT MCQ

We can write (x - y)² as (x - y) * (x - y) = x² - xy - yx + y²
= x² - 2xy + y².

As per the observation made from the polynomial 7x² + 6x - 2, x is being multiplied by the number 6. Therefore, it is hereby deduced that 6 is the coefficient of x in the polynomial 7x² + 6x - 2. Furthermore, 7 cannot be the coefficient of x as it is being multiplied by x². 

We can get number of terms by separating them by “+” or “–” sign.
In the given polynomial 5x² + 2x - 2, if we separate expressions by “+” or “–” sign, we get three separate terms as 5x², 2x and 2.
Hence we can say that there are three terms in the polynomial 5x² + 2x - 2.

p(x) = -5x + 5
-5x+5 = 0
5 = 5x
x = 1

Polynomials only having three terms are called Trinomials.
Hence to find trinomial polynomial, we have to find polynomial which has three terms. We can see that 5x² + 2x - 2 has three terms namely 5x², 2x and 2. Hence it is trinomial.
Similarly, polynomials only having one term are called Monomials and polynomials only having two terms are called Binomials.

If p(x) = cx + d and cx + d = 0, then cx = -d. Furthermore, the outcome is x = -d/c. Therefore, it is derived that the zero for the polynomial p(x) = cx + d is -d/c, in accordance with the value of x which is -d/c. 

If t = 0, then p(t) = 2 + t + 2t² + t³ would be p(0) = 2 + 0 + 2(0)² + (0)². Its further results in 2. Therefore, according to the derivation, an observation can be made that the value of p(t) = 2 + t + 2t² + t³ would be 2 when t = 0.

Any polynomial with two terms along with degree 20 is a binomial of degree 20. As per the observation made from the polynomial x²⁰ + 1, the degree is one as it is the highest level in terms of variables. Furthermore, the polynomial has two terms x²⁰ and 1. 

Since 3 = 3x0, 3 is to be a constant polynomial. On the other hand, 4x + 1 and 6x + 3 are linear polynomials. Furthermore, 2x² is a quadratic polynomial. 

Any polynomial bearing degree one is a linear polynomial. As per the observation made from the polynomial 1 + 3x, the degree is one as it is the highest level in terms of variables. Therefore, the type of polynomial is linear in nature. 

If f(x) = 2x + 7 and 2x + 7 = 0, then 2x = -7. Furthermore, the outcome is x = -7/2. Therefore, it is derived that the zero for the polynomial f(x) = 2x + 7 is -7/2, in accordance with the value of x which is -7/2. 

If x = -1, then f(x) = 5x − 4x² + 3 would be f(-1) = 5(-1) − 4(-1)² + 3. It further results in -5 - 4 + 3. Further calculation leads to the outcome as -6. Therefore, according to the derivation, an observation can be made that the value of f(x) = 5x − 4x² + 3 would be -6 when x = -1.

Since the degree at the highest level for the polynomial 2x² + x - 1 is two according to variables, it is to be a quadratic polynomial. On the other hand, 7x + 3 is a linear polynomial. Furthermore, x + 3x³ + 1 is a polynomial that is cubic

72 would mean 7 needs to be multiplied by 7. The outcome will be 49. 52 would mean 5 needs to be multiplied by 5. The outcome will be 24. Therefore, if 25 is subtracted from 49, the outcome will be 24. As a result, it is hereby deduced that 72 - 52 will be 24.

Any polynomial bearing degree three is a cubic polynomial. As per the observation made from the polynomial x – x³, the degree is three as it is the highest level in terms of variables. Therefore, the type of polynomial is cubic.